MULTIPHASE AVERAGING OF PERIODIC SOLITON EQUATIONS
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University Microfilms International 300 N. ZEEB ROAD, ANN ARB08, Ml 48106 18 BEDFORD ROW, LONDON WCIR 4EJ, ENGLAND 8003057
FOREST, MARK GREGORY
MULTIPHASE AVERAGING OF PERIODIC SOLITON EQUATIONS
The University of Arizona PH.D. 1979
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Universi^ Micrailms International 300 N. 2EEB RD.. ANN ARBOR. Ml .18106 (3131 761-4700 MULTIPHASE AVERAGING OF PERIODIC
SOLITON EQUATIONS
by
Mark Gregory Forest
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF MATHEMATICS
In Partial Fulfillment of the Requirements For the Degree of
DOCTOR OF PHILOSOPHY
In the Graduate College
THE UlTIVERSITY OF ARIZONA
19 7 9 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
I hereby recommend that this dissertation prepared under my direction
by M. Gregory Forest
entitled "Multiphase Averaging of Periodic Soliton Equations"
be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy
h/iAnl td ykhujAUM^ AIM, X ^ H79 Dissertation Director / Date Q J
As members of the Final Examination Committee, we certify that we have read this dissertation and agree that it may be presented for final defense.
najn.! li) Hc-k-MAki jA iLAX.
Date ^ A , lo , I
7979
Date
Final approval and acceptance of this dissertation is contingent on the candidate's adec[uate performance and defense thereof at the final oral examination.
11/78 STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under the rules of the library.
Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotations from or reproduction of the manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of this material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.
SIGNED 1^- 0 I This dissertation is dedicated to the woman of my life,
Barbara, my son, Scott, Mom and Dad, Lee, Robin and her growing family, the Perraults, the Phillips, and the Harahan Gang.
iii ACKNOWLEDGMENTS
My sincere thanks goes to my adviser, De.v^ McLaughlin.
As a person, his patience helped me suirvive the bad times, and as a scholar, he shared with me his thirst and wealth of knowledge.
I also express gratitude to Hermann Flaschka and George Lamb, from whom I have gained much more than they realize.
iv TABLE OF CONTENTS
Page
LIST OF TABLES viii
LIST OF ILLUSTRATIONS ix
ABSTRACT xi
CHAPTER
I. SPECTRAL THEORY FOR THE PERIODIC SINE-GORDON EQUATION 1
l.I Introduction 1 1.II Preliminaries 2 1.11.1 Sine-Gordon Spectral Theory; Analogies Between Periodic and Whole-Line Problems 2 1.11.2 Transfer Matrix, Floquet Discriminant, and Floquet Solutions 6 1.11.3 Generic Properties of the Spectrum 0 9 l.III Qualitative Insight into the Spectrim a 14 l.IV Finite Degrees of Freedom 27 l.IV.l 2N Invariant Simple Spectra {E,}; Determination of a Riemann Surface 27 l.IV.2 i^-Representation of the N-Phase Wave Train. . . 30 l.IV.3 0-function Representation of u(x,t); N-Phase Wave Train 37 l.V Traveling-Wave Solutions (Single-Phase Wave Trains). . 41 l.V.l Reduction to N = 1 41 l.V.2 Direct Ansatz Method to Display Traveling Waves.41 l.V.3 p-Representation of u(x,t) for N = 1; Catalog and Physical Characteristics of the Traveling Waves In Terms of = {E^,E2} 47 l.V.4 Infinite-Product Representation of 6-Functions and Simmiation Representation of the Traveling Waves 57 l.VI Separable Solutions (N = 2) 73 l.VI.l Definitions and Motivation 73 l.VI.2 Even Initial Data (u(x),II(x)) <=> "Open Circuit" Boundary Conditions 75 l.VI.3 Spatial Symmetry <=> Spectral Symmetry 76
V vi
TABLE OF CONTENTS—Continued
1.VI.4 Consequences of the Spectral Synmetry (N = 2); Separability, Standing Waves, and Elliptic Functions 78
2. CANONICAL VARIABLES FOR THE PERIODIC SINE-GORDON EQUATION AND A METHOD OF AVERAGING 86
2.1 Introduction 86 2.II A Prescription for N-Phase Modulational Equations of Completely Integrable Hamiltonian Systems 88 2.Ill Remarks on the Validity of the Modulational Equations. 95 2.IV Canonical Variables for the Periodic Sine-Gordon Equation 103 2.V Reduction to Finite (N) Degrees of Freedom 106 2.VI Action-Angle Description for N Degrees of Freedom. . . 110 2.VII Method of Averaging : N = 1 Case 11A 2.VIII Future Considerations 121
3. MULTIPHASE AVERAGING AND THE INVERSE SPECTRAL SOLUTION OF THE KORTEWEG-DEVRIES EQUATION 122
3.1 Introduction 122 3.II Whitham's Single-Phase KdV Results 122 3.Ill N-Phase KdV Waves and the 1ST Interpretation of Whitham's Single-Phase Results 133 3.IV 1ST Approach to Multiphase Averaging of the KdV Equation 139 3.IV.I KdV as a Completely Integrable Hamiltonian System and Conservation Laws 139 3.IV.2 Simplification of the Averaging Integrals; Ergodic Flow on an N-Torus 142 3.IV.3 The Role of Abelian Differentials in the Modulational Equations; Summary of Results. . .144
4 CONSEQUENCES OF THE INVARIANT REPRESENTATION Ji = 0 148
4.1 Introduction 148 4.II = 0: Frequencies, Wave Numbers, and Conservation of Waves 148 4.Ill ^ and the Action Integrals 150 4.IV = 0 and the Canonical Variables Approach to N-Phase Averag ing 153 4.V Concluding Remarks 159 vii
TABLE OF CONTENTS—Continued
Page
APPENDIX A: SCATTERING MOTIVATION 161
APPENDIX B: DERIVATION OF THE DECOMPOSITION FORMULAS 168
APPENDIX C: PROOF: SPATIAL SYMMETRY <=> SPECTRAL SYMMETRY 170
APPENDIX D: PROOF OF THEOREM 2.IV.1 175
APPENDIX E: CONNECTION BETWEEN DATE y VARIABLES AND THE "y SPECTRUM" ^ 183
APPENDIX F: N DEGREES OF FREEDON; DERIVATIONS 187
APPENDIX G: RIEMANN BILINEAR IDENTITIES. . 201
APPENDIX H; MULTIPHASE AVERAGING AND THE INVERSE SPECTRAL SOLUTION OF K.dV 208
REFERENCES. 278 LIST OF TABLES
Table Page
1. General Structure of the Loop Integrals 1(a), 1(b), Normalization Constant C, and Period Matrix B...... 68
2. Elliptic Integral Formulas for the Normalization Constant C and Period Matrix B 69
3. Soliton Limit (E^ = E2 < 0) Formulas 70
4. Contrasts in Physical Characteristics of the Oscillatory States in Terms of E^,E^ 71
viii LIST OF ILLUSTRATIONS
Figure Page
1. Truncated Potentials 15
2. Spectrum a for Whole-Line Problem with Vanishing Boundary Conditions 17
2^. Whole-Line Spectrum with Vanishing Boundary Conditions. . . .18
3. Graph of A(E) vs. E real 21
4. Spines in the Floquet Spectrum a(>i) 22
4^. Spines in the Floquet Spectrum a(E) 22
5. Generic Profile of the Floquet Spectrum ct(X) 25
5^. Generic Profile of the Floquet Spectrum a(E) 26
6. Sample Cut Structure for Genus N = 3 Riemann Surface 30
7. a,b-cycles for Genus N = 3 38
8. Potential Energy Diagrams and Corresponding Solutions (N = 1) 43
9. Various Locations of E,,E„ with Appropriate Branch Cuts (N = 1) 47
10. Graph of U vs. (E^E2)^ 50
11. y-cycle; N = 1, E^ < E2 < 0 53
12. y-cycle; N = 1, E^ = E*, 5^ E2 55
13. Pure Oscillator (|u| < 1); Effective Oscillator Diagram. . . 56
14. Familiar Kink, Antikink Shapes 58
15. Graphic Representation of the Decomposition Formula for the Kink Train, E^ < E2 < 0 62
16. Graphic Representation of the Decomposition Formula for the Oscillatory State, E^ = E^, E^ 7^ E2 62
ix X
LIST OF ILLUSTRATIONS—Continuted
Figure
17. Analyticity Structure for the Soliton Limit, = Eg < 0. . 63
18. Soliton Limit, E^ = E* < 0. .• 65
19. Canonical a,b-cycles and Useful Contours of Integration for All N = 1 Cut Structures 67
20. Graphic Limit as ph E^ : ir ^ 0, |e^1 constant 72
21. Branch Cut Structures for Two-Phase Solutions With Open Circuit Boundary Conditions. 80
22. Image of the Circle of Symmetry [E] = 85
23. Blocks on the X (long) Scale 91
24. Potential Energy Diagram 125
25. Band Structure of a and "y-cycles", N = 4 135 ABSTRACT
The multiphase averaging of periodic soliton equations is
considered. Particular attention is given to the periodic sine-
Gordon and Korteweg-deVries (KdV) equations. The periodic sine-
Gordon equation and its associated inverse spectral theory is
analyzed, including a discussion of the spectral representations
of exact, N-phase sine-Gordon solutions. The emphasis is on
physical characteristics of the periodic waves, with a motivation
from the well-known whole-line solitons. A canonical Hamiltonian
approach for the modulational theory of N-phase waves is prescribed.
A concrete illustration of this averaging method is provided with
the periodic sine-Gordon equation; explicit averaging results are
given only for the N = 1 case, laying a foundation for a more
thorough treatment of the general N-phase problem. For the KdV
equation, very general results are given for multiphase averaging
of the N-phase waves. The single-phase results of Whitham are
extended to general N phases, and more importantly, an invariant
representation in terms of Abelian differentials on a Riemann
surface is provided. Several consequences of this invariant represen
tation are deduced, including strong evidence for the Hamiltonian structure of N-phase modulational equations.
xi CHAPTER 1
SPECTRAL THEORY FOR THE PERIODIC SINE-GORDON EQUATION
l.I Introduction
This chapter is devoted to a useful understanding of the
periodic sine-Gordon equation and its associated inverse spectral
theory (1ST). Very general exact solutions of the periodic sine-
Gordon equation have been developed through 1ST [9,30], but to be
useful from the point of view of applications, explicit connections
between the parameters in the 1ST representation and the physical
characteristics of the wave are necessary. Throughout this paper,
we motivate the periodic theory with the more familiar whole-line
sine-Gordon equation, its celebrated soliton solutions (such as kinks, antikinks, breathers), and the role of inverse scattering
theory in the exact integration of the sine-Gordon equation. In
particular, we seek (i) a classification of the fundamental excita tions in the periodic sine-Gordon field based on the location of the
periodic spectrum; (ii) the relation between the periodic field configurations and the whole-line solitons; (iii) a feeling for the generic structure of the periodic spectrum; and (iv) a quantitative connection between the 1ST input parameters and the physical characteristics of the periodic sine-Gordon wave.
1 2
l.II Preliminaries
1.11,1 Sine-Gordon Spectral Theory; Analogies Between Periodic and Whole-Line Problems
Consider the sine-Gordon equation in laboratory coordinates,^
u^. - u + sin u = 0, -oo The key to the integration of all nonlinear soliton equations is to adjoin to the nonlinear partial differential equation a linear eigen value problem whose "potential" is given by the solution of the soliton equation. Just as the linear SchrOdinger equation (-i|) +u(x) i() = E,) can be used to integrate the nonlinear Korteweg-deVries equation [18] (u - 6uu + u = 0), the integration of the sine-Gordon equation is intimately related to the following linear eigenvalue problem: xu 0 -1\ , . /O 1' - /E ^ = "5. (l.II.2) ^ + 7" -iu 1 o; '' 11 0, \0 LV where w e u^ + u^, ^ Indeed, the linear system lU 'o -1^ 'o 1^ d 1 T~ + 7" W - ^ =15 (1.11.3a) dx A -iu L\ 0 1 0 16/E \0 lU 0 -1^ 0 d , 1 •JZ + T W - /e ^ = d, (1.11.3b) dt 4 a 0/ 16/E \O E-^"I ^ In the entire paper, x,t subscripts denote partial derivatives. 3 which consists of the linear problem (1.II.2) augmented by a time flow for is compatible (^ ^. = ) if and only if the potentials u and w satisfy a system which is equivalent to the sine-Gordon equation, u^ + u = w w^ - w^ = - sin u. In this manner, the linear system (l,II.3a,b) implicitly carries the content of the sine-Gordon field as a compatibility condition. This method for the integration of the sine-Gordon equation is due to Lamb [26] and has been clarified and developed by Ablowitz, Kaup, and Newell [3] and Takhatajian and Faddeev [39] . Since we use the exposition of the latter authors, we refer to equation (1.II.2) as the "Takhatajian-Faddeev eigenvalue problem." That the sine-Gordon equation (l.II.l) arises as a compatibility condition for the system (l.II.3a,b) is a fact which is local in Cx,t), independent of boundary conditions. Once boundary conditions are imposed upon the sine-Gordon equation, the potentials u and w of the Takhatajian-Faddeev eigenvalue problem (1,11,2) inherit the same boundary conditions. This boundary behavior of the potentials fixes the type of spectrum which the eigenvalue problem Cl.II-2) possesses. The nature of the spectrimi, in turn, is intimately connected to the classes of fundamental excitations of the sine-Gordon field under the particular boundary conditions. For example, consider the sine-Gordon equation under vanishing boundary conditions at Ix| = »; - u + sin u = 0, -oo u(x,t) ^ 0 (mod 2ir) as Ix| ->• <*>, (1,11.4b) u^(x,t)->0 as Ix| ->•«>. (l.II.4c) It is well known that the fundamental excitations in this case consist of solitons (kinks), anti-solitons (antikinks), breathers, and radiation. (See Figure 14, p. 58 arid [36] for a detailed discussion.) When the potentials u(x) and w(x) of the Takhatajian-Faddeev eigenvalue problem (1.II.2) satisfy these vanishing boundary conditions (l,II.4b,c), this eigenvalue problem, viewed over the entire x-axis, has continuous spectrum filling the positive E-axis, and discrete bound states (eigenvalues) which occur either at points on the negative real E-axis or in complex conjugate pairs (see Figure 2^). The continuous spectrum for E > 0 is related to radiation degrees of freedom for the sine-Gordon field, the bound state eigenvalues E^ < 0 label kink or antikink solutions, and the conjugate bound state pairs E^, Et label breather excitations. As the potentials u,w flow iti time according to the siner^Gordon equation, the locations of the bound state eigenvalues remain fixed and determine the speeds and widths of the solitons and the frequencies of the breathers. This invariance of the spectrum in time is central to the integration of the sine^Gordon equation, which is said to '^generate an isospectral flow"' for the Takhatajian-Faddeev eigenvalue problem. 5 In this paper, we are conceimed with the sine-Gordon equation under periodic boundary conditions, with period L; u^. - u + sin u = 0, -o> u(x + L,t) = u(x,t) (mod Ztt) (1.11.5b) u^(x + L,t) = u^(x,t). (1.11.5c) When the potentials u and w of (1.II.2) satisfy these boundary conditions (l.II.5bc), the entire spectrtm of (1,11,2), still defined over the whole x-axis, is continuous spectrum. This spectrum consists of curves (or bands) which can be interpreted as follows; (i) the discrete bound states for potentials with vanishing boundary conditions (l.II.4bc) have spread into narrow bands of continuous spectrum, and (ii) short "spines" of. spectrum have grown off the positive E-axis (see Figure ( 4'')) • The basic excitations of the periodic sine-Gordon field then consist of periodic trains of kinks or antikinks (associated with the narrow bands of spectrum on the negative E-axis), trains of breathers (associated with the narrow bands in conjugate pairs), and radiation which behaves much as Fourier modes for the linear Klein-Gordon field (and is associated with the spines off the positive real axis). The isospectrality is now manifested by the invariance of these bands of spectrum under the periodic sine-Gordon flow. This spectral classification of the fundamental excitations in the periodic sine-Gordon field will be established in the following sections. We remark that, to be useful in applications, the correspondence between the location of the spectrum and physical characteristics of the potentials must be concretely understood; some of these facts are provided in the body of the paper. With these remarks in mind, we view the Takhatajian-Faddeev eigenvalue problem (1.II.2) as a family of eigenvalues problems, defined on the entire real x-axis. This family, indexed by a time parameter t, is generated by the periodic sine-Gordon equation (l.ir.5). Since the spectrum a is Invariant in t, a is determined o ° for all time by the periodic initial data u, 11, o o u(x,t=o) H u(x), u^(x,t=o)= n(x). o o u(x+L) = u(x) (mod 2ir) o o n(x+'L) = n(x). via o iu '0 -1\ i o o /O 1\ ^ + T (n + u ) I 1 + .o -/e ^ =•$. (1.II.6) il 0/ ^ \l 01 16v^ \0 e~^" We begin with a discussion of the general properties of the spectrum a, o ° and later consider special cases of u, II. 1.II.2 Transfer Matrix, Floquet Discriminant, and Floquet Solutions First, we define the tools which are used to display the spectrum a, namely, the transfer matrix, Floquet discriminant, and Floquet solutions. Fix a point x^, and a basis of solutions of (1.II.6), {^_j_(x,x^,E), ^_(x,x^E)}, by the following initial conditions at x = x : o '1^ 'O ^ |'^(x=x^,x^,E) = , |"_(x=x^,x^,E) = 10/ U/ Notice that (|)_j^(xfL,x^,E) are also solutions of (1.II.6) given the solutions ^_j_(x,x^,E); this follows quickly from the periodicity property of w, exp(+iu) . Therefore, we can expand these "new" solutions on the basis ^_j^(x,x^,E): (j)^(x+L,x^,E) = t^j^(E) ->^(x,x^,E) + $_(x,x^,E) ^_(x+L,x^,E) = t22(E) ^^(x,x^,E) + t22(E) ^_(x,x^,E), or more concisely. ^^(xfL,x^,E) |'_^(x,Xp,E) = T(E) (1.II.7) ^ (x+-L,x ,E) |'_(x,x^,E) The (2x2) "transfer matrix" T(E) transfers the basis ^_^(x,x^,E) across one period L of the potentials u,w. Iterating this formula (1.II.5) across N periods yields y , VATiMij « A. o ' N = T"(E) (1.II.8) ^_(x+NL,x ,E)y ^_(x,x^,E) 8 By definition, a complex number E belongs to the spectrum a of the Takhatajian-Faddeev eigenvalue problem (1.II.6) if and only if the solutions (^^(x,x^,E) are bounded for all x, or equivalently, if and only if ^_j_(x+NL,x^,E) are bounded for all integers N. From (1.II.8), we see that such bounded behavior for large N is possible if and only if both eigenvalues, p_^(E), of the transfer matrix T(E) have unit modulus. Thus we obtain Fact 1; E € 0 if and only if Ip^(E)I = 1, where p^(E) are the eigenvalues of the transfer matrix defined as the two roots of det[T(E) - p(E) I] = 0. Computing this determinant yields p^(E) - A(E) p(E) + 1 = 0, (1.II.9) where A(E) = trace of T(E). The roots are given explicitly by . A (E)+i^^^7E)^ P+(E) (I.II.IO) The function A(E) is known as the Floquet discriminant, and is central to the entire theory. From (1.II.9), (1.11.10) it follows P_(E) • P+(E) = 1, A(E) = p_(E) + p^(E), so that 1P^(E)| = 1 if and only if A(51 i^ ^^ea,lp with. |A(E)| 2. We therefore have Fact 2; E 6 a iff A(E) is real, with |A(E)| £ 2. We remind the reader that the eigenvalue problem (1.II.6) is not self-adjoint, so that its spectrum a need not be real. Nevertheless, a does lie on curves of real A(E), Next, we change basis from (x,x^,E)} to one in which the transfer matrix T(E) becomes diagonal. This new basis is called the Floquet basis and consists of the Floquet solutions. {T^i^^(x,x^,E)}, defined by (x+L,x^,E) = (E) (x,x^,E). (l.II The eigenvalues p^(E) are called the Floquet multipliers; the representation (l.II.ll) clearly indicates that Floquet theory is a natural generalization of strictly periodic functions (p=l) and can be used to develop generic properties of the spectrum 0. l.II.3 Generic Properties of the Spectrum a Using these ingredients (transfer matrix T(E), Floquet discriminant A(E) , Floquet multipliers p^(E), ELoquet basis , we can sinnmarize the generic properties of the spectrum a in Theorem l.II.l (Useful Properties of the Spectrum g) (i) E € a iff ^_j_(x,x^,E), I|I^^(x,x^,E) are bounded for all x £ iff IP(E)| = 1 iff A(E) is real and |A(E)| 2. 10 (ii) The spectrum a is continuous spectrum. It consists of a countable number of smooth curves, called bands of spectrum; the endpoints of these curves are simple periodic and antiperiodic eigenvalues. (iii) E is a periodic (antiperiodic) eigenvalue iff A(E) = + 2 (-2). (iv) For real potentials u and w, the following symmetries hold; If E € a, then E € a. A If E is a periodic eigenvalue, then E is a periodic eigenvalue. * If E is an antiperiodic eigenvalue, then E is an antiperiodic eigenvalue. Proof of the Theorem: Part (i) is clear from Section 1.II.2. However, it is instructive to iterate formula (l.II.ll) N periods to the left and right; T^^(X+NL,X^,E) = p^(E) iji^Cx.x^jE) (Tl) i|i^(x-NL,x^,E) = p~^(E) ^_^(x,x^,E). (T2) It then becomes obvious that for |p_|_| ^ 1, the corresponding eigenfunctions must blow up either at x = +'» or x = -®. Otherwise, for |p_j_| = 1, (Tl) and (T2) show the eigenfunctions remain bounded for all x. Since A(E) = p,(E)+p (E)=p,(E)H 7^ + - -f- P^.v.ii; ^ ( ) refers to complex conjugation here and in the remainder of the paper. 11 ±t immediately follows from |p_j_| = 1 that A is real and IA CE)| ^ 2, Part (i) is thereby proved. To prove Part (iii), we use (Tl) with N = 1. Clearly p= + l(-l) yields periodic (antiperiodic) Floquet solutions; since p = "*"2 ^ , A(E) = +2(-2) determines these periodic (antiperiodic) eigenvalues. For part (ii), since a is characterized by |p| = 1, from CTl) and (T2) we see that Xispaats the same values over each period L. It follows that w-: le bounded for all x, can not vanish as [x] -»•<»; thus are generalized eigen- functions and a is continuous spectrum. To see that a lies in bands, we use the basic fact, without proof, that A(E) is an analytic function of E except for essential singularities at E = 0, ». (See the Remark at the end of this section.) By Part (i), a is characterized by A(E) real, |A(E)| ^ 2; A(E) analytic then yields that a lies on smooth curves of Im(A(E)) = 0, terminating only at these points where A(E) = + 2, A^(E) ={= 0« that is, the simple (anti) periodic eigenvalues. Thus Part (ii) is established. / \ ^ (x,x ,E) \ To establish Part Civ), let i^(x,x ,E) - , , denote a Floquet solution of the eigenvalue problem (1.II.6) at E € a. Simply by plugging into (1.II.6), it follows that 12 i(/*(x,x ,E)\ ^ ^ is a Floquet solution at E if we assume reality of u(x) and w(x). From the hypothesis E 6 a, and likewise \p^, - are bounded; E € a. The periodicity (anti) follows immediately upon inspection. I_| We close this section with a technical representation of the discriminant A(E) which is extremely useful for displaying qualita tive feature, of the spectrum a. In the basis {^+(x,x^,E)}, we have the following Theorem 1.II.2 (Eigenfunction Representation of A(E)) The Floquet discriminant A(E) can be represented as A(E) = (l)_j^^l(Xo+L,Xo,E) + (1.II.12) Proof: In this basis {^_j_(x,x^,E)}, we have (^, (x+L,x ,E)\ ''|'^(x,x^,E)' ^ ° = T(E) ^|^_(x+L,x^,E)y \,^_(x,x ,E), Setting X = x^ and using the initial conditions for at X = x^ yields ^^(x^+L,x^,E) = t^^(E) + 42(^5 (J I.Cx^+L.x^.E) = t2i(E) |jj + t22(E) from which we find T(E) = V'i'_,1 (*-, 2 (»E) ^ Since A(E) = trace of T(E), we have (1.II.12). [ Remark; Since ^^(x,x^,E) satisfy an initial-value problem with (o)' (l) data, it can be established by a Picard iterate scheme that (j), T (x,x ,E) and (j) «(x,x ,E) (but not at E = 0, " (check, for example, the special case u = w = 0). Hence, equation (1.II.12) yields that the Floquet discriminant A(E) enjoys the same analyticity properties. 14 l.III Qualitative Insight into the Spectrum a The purpose of Sections l.II, l.III of this paper is to connect properties of the spectrum a with the solution of the periodic initial value problem for the Sine-Gordon equation. Thus far, in Section l.II we have discussed the general nature of the spectrum o; here we seek more detailed, qualitative information about the structure of the bands of a. To obtain this information, we imbed a whole-line scattering problem into each period. Thereby, we provide simple derivations of the properties of the Floquet spectrum a based upon better known-properties from whole-line scattering theory; we then interpret the relation of this generic band structure of a to solutions of the sine-Gordon equation. With these remarks in mind, we specialize to potentials u(x), such that u(x) has compact support (mod 2ir) within each period; 0 Il(x) has compact support within each period. We call such potentials truncated potentials. (See Figure 1.) Although our derivations are restricted to this class of truncated potentials, the specific properties which we obtain are in fact more general. We emphasize that this class contains potentials which are far from the vacuum (8=5?^ 0); thus, this representation gives exact spectral information about sine-Gordon field configurations which contain arbitrary numbers of the basic excitations (periodic trains of kinks and antikinks, breathers and radiation), For convenience, we summarize useful whole-line scattering properties in Appendix A. The ordering for the remainder of Section l.III is to first use infinite-line scattering theory to arrive at a "Scattering Representation" of the discriminant A(,E), from which we deduce the general qualitative structure, of the spectrum; finally, we relate this structure to the fundamental excitations of the sine-Gordon field. Consider truncated potentials u(.x), w(x) within the period [x^,x^ + L] (Figure 1), for which we have an extremely useful repre sentation of the Floquet discriminant. Let X = /e. w(x) X +L S(x) ^(x) Figure 1 Truncated Potentials 16 Theorem l.III.l (Scattering Representation of the Floquet Discriminant A(E)) Let u(x) and w(x) denote truncated potentials. Then, o 2 in terms of the scattering parameter^ X, X = E, A(X) = a(X)e"^°'^^^^ + aTxTe^"^^^^, ImX > 0. (l.III.l) Here a(X) h X - , [a(X)] ^ denote the transmission coefficient across one period of the potentials, and the potentials satisfy u(x) = O(mod 2-n), w(x) = 0 outside + L]* For X real, with a(X) = |a(X) ] A(X) = 2la(X)I cos[a(X)L - ph a(X)], X real. (1.III.2) ^As will become evident in the next section, the appropriate mathematical function theory takes place on a Riemann surface built from two copies of the E-plane, with branch cuts along the continuous spectrum of the Takhatajian-Faddeev eigenvalue problem. In whole- line scattering theory, this spectrum consists of the positive real E-axis, and thus the branch cut structure coincides with that of the function X^=E. However, for periodic problems the continuous spectrum is more complicated; it consists of bands of spectrum with endpoints {Ej}. Thus the branch cuts for the periodic problem coincide with that of the function X^=E ^(E-E^). Therefore for whole-line scattering theory, X=E^ is a natural local coordinate (except near "); however, for periodic problems it is not a natural local coordinate (except near 0). Nevertheless, for the remainder of this section we will work with the parameter X since we aim to use facts from whole-line scattering theory. We present the proof of Theorem l.III.l in Appendix A so as not to delay the exposition. In these representations (l.III.l), (1.III.2) of A(X), a(X) satisfies the following properties from whole-line scattering theory (see Appendix A). Theorem 1.III.2 (Facts From Whole-Line Scattering Theory.) (i) [a(X)] ^ = transmission coefficient (ii) as X -»-oo, ImX > 0 a(X) -»• exp[-i(u(x^+L)-u(x^))] as X-^0 (iii) |a(X)|^ + |b(X)|^ = 1, X real (iv) a(X^) =0 if and only if X^ is a bound state eigenvalue. These bound states occur either on the positi\'e imaginary X-axis (in which case they are associated with solitons, i.e. kinks or antikinks) or in pairs (X^j-X^), ImX^ > 0 (in which case they are associated with breathers). We summarize these properties in Figure 2 below. fX^ '^o'^l'^2~^ kinks or antikinks -X* ^3 (Xgj-Xg) breather Xii real X => radiation X-plane Figure 2 Spectrum o for Whole-line Problem with Vanishing Boundary Conditions 18 The analogous picture in the E-plane is given in Figure 2 , where E. = X. 1 X E^,Ei,E2 => Kinks (Antikinks) Eg, Eg Breather .^3 E > 0 => Radiation El E2 • E„ E-plane Figure 2^ Whole-line Spectrum with Vanishing Boundary Conditions We now deduce a series of facts based on the scattering representations of A given in Theorem l.III.l; we begin with Fact 1; The entire real X-axis is continuous spectrum; thus there are no gaps in the spectrum on the real X-axis. In the E-plane this implies Fact 1^: The positive real E-axis (E > 0) is continuous spectrum, with no gaps in a on the positive real E-axis. This part of the spectrum is associated with radiation in the sine-Gordon field. These facts are fairly obvious from the following formulas (see Theorems l.III.l, 1.III.2); for X real |a(X)l^ + |b(X) ] ^ = 1 |a(X) ]£ 1 =» 1a| = 2la(X)| Icos[. ]| £2 Since A is real and always bounded in magnitude by 2, the entire real X-axis is continuous spectrum (Theorem l.II.l). 19 Next, consider the special case uCx) = wCx) = 0. In this case of zero potentials the scattering representations of A together with the total transmission (a(X) = aCX) E 1) property of zero potentials give: For u(x) = w(x) E 0, with a(X) = » A(X) = ImX 0, A(X) = 2cosh[|x| + 25!^I^^ imaginary, ImX > 0. A(X) = 2cos[a(X)L], X real. From these equations we deduce Fact 2; For zero potentials, u(x) = w(x) = 0, (a) For ImX ^ 0, [A(X)[ » exponentially fast as ImX ^ «>. (b) For X pure imaginary, ImX > 0, A(X) is real and |A(X)| ^ » exponentially fast as |x| -»• 0 or <*>. (c) For X real, the oscillations always reach A(X) =+2; all roots of A(X) = + 2 are double; A(X)~2cos(XL) as X ->• «*> so the graph has a very regular appearance, A(X)'»2COS(-Y^) as X -»• 0 and the graph appears very ^16X' dense. In terms of the eigenvalue parameter E, X 2 = E, we have (see Figure 3) Fact 2 ; For u(x) = w(x) = 0, (a) |A(E)[ 00 exponentially fast as |E| -> », E [0,«>). (b) For E < 0, A(E) is real with A(E) -v ® exponentially fast as E 0,". 20 Cc) For E > 0, the oscillations of A(E) always reach + 2; all periodic and antiperiodic eigenvalues are double; A(E) ~ 2cos L) as £->• + <», and the graph appears very regular near E = + », A(E)'«'2cos(—as E -»• o"^, and the graph appears 16/e very dense near E = 0, E > 0, The general case, u, w ^ 0, asymptotes to these formulas as E -»• 0,", with the following exception near E = 0. First, there is a possible phase shift of ir as E->-0,E>0 (with the "charge" n defined u(x^+L) = u(x) + 2n'n', the phase shift is zero for n even and ir for n odd). That is, for general potentials u,w, A(E)^(-1)^ 2cos(—near E = 0, E > 0, 16v^ A (E)^ 2cos(v^ L) near E = + », E real. Second is the effect of the "charge" of u(x) on the graph near E = 0, E < 0: +" n even lim A(E) =< E->0~ ] n odd With these remarks, we can sketch A(E) vs. E real (Figure 3). 21 A = +2 »1 -2 a u(x +L) = u(x ) + 2mr, n even III + > to 1 1] •••« / I / ' 1 / "" / 1 1 / / 1 ] / \ h \ 1 E I / ° 1 / 1 III M > 1 b. u(x +L) = u(x ) + 2mT, n odd o o ' Figure 3 Graph of A(E) vs. E real In these developments, we have found the entire positive real E-axis Cor real X-axis) is continuous spectrum. There is an interesting ramification of this fact: Fact 3; In the X-plane, the spectrum a has spines coming off the real X-axis (see Figure 4). Fact 3 ; In the E-plane, the spectrum a has spines coming off the positive real E-axis Csee Figure , X-plane Figure 4 Spines in the Floquet Spectrum a(X) E-plane Figure 4'' Spines in the Floquet Spectrum a(E) 23 The proof of Fact 3 , a rather surprising result in sine- Gordon spectral theory, is really just an elementary exercise in complex function theory.*^ The gist of the argimient is as follows. Refer to the graph of A(E) vs. E real, E > 0 (Figure 3), and consider a local maximum (E^, A(E^)) where |A(E^)| < 2 and (E^) = 0. (Remember A(E) is real for E real and o lies on curves of real A(E), that is, curves of ImA(E) = 0, with |a1 < 2.) From a more general perspective, though, A(E) E Aj^(Ej^,Ej) + i is an analytic function of the complex variable E = E^^ + iE^; thus harmonic so E^ must be a saddle point on the surface z = A„(E_,ET). Thus, leaving E K K J- O along the real E-axis one descends the saddle along a curve of steepest descent. But since E^ is a saddle point, there is another direction off the real axis into the complex E-plane which is a steepest ascent path. (In fact, the curve is determined by A^EO.) Then by continuity (since |A(E^)1 < 2), this curve gives a short band of spectrum, a spine. Clearly the same holds true at a local minimimi E^ where |A(E^)1 < 2.^ 1_1 As E -»• 0, + <*>, the behavior of A(E) shows the length of these spines must asymptote to zero. They tend to be very short anyway; this seems intuitively clear from the above arguments since a spine is a steepest descent (ascent) path of A^ and so should IS. ^ The proof follows almost verbatim the arguments for the steepest descent method of asymptotics [6]. ^ We emphasize that this proof is an analyticity argument, independent of the compact support restrictions inherent in the scattering representation of A. 24 exceed 2 in magnitude very quickly. In fact, equation (l.III.l) for A(X), ImX ^ 0, shows |A(X)| grows exponentially as a function of Xj ^ 0; A(X) = a(X)e~^"^^^^ + a(X)e^^''^^ , X = X^^ + iX^ , i[phaCX)- X (1- ygi^)L] XjL(l = |a(X)|e ' ' e ' ' 1 s,n , w, . 1 i[pha(X) + Xjd - x^)L] -XjLd + jjpp.) + |a(l)|e ' ' e "i i _ (1.III.3) from which we have X^L(1 + IA(X)I |a(X)|e , Xj > 0. (1.III.4) This completes our discussion of the spectrum a which is connected to the real X-axis (or positive real E-axis). There also exist parts of the spectrum which do not lie on the real X-axis (positive real E-axis). These are the periodic generalizations of kinks (antikinks) and breathers. From the exponential growth of A(X) for X^. > 0, equation (1.III.4), we realize any such bands of spectrum in the upper half X-plane, or off the positive real E-axis, are very short and sparse. They originate from a different type of oscillation in A(X), coming from zeros of a(X ). Assimie we have some curve A^ = 0 in the upper half X-plane. If there is to be any spectrum along such a curve. 25 we must have -2 ^ A < 2, However, from the exponential behavior K of A(X) for Xj > 0, X^LCl + LETT] IACX) I |a(X)|e » ^ P' (1.TII.5) A = A, will usually exceed 2 unless a(X) has a zero very near R' the curve. The zeros of a(X) are finite in nimber and occur only at the bound states of the truncated potentials u(x), w(,x), Since these zeros of a(X) are isolated, |A(X)| will exceed 2 very quickly. Thus, there will be a short band of spectrum whenever the zeros of a(X) occur sufficiently nearby a level curve Aj. = 0. These facts, together with the sjmmietries of Theorem (l.II.l), E € a # E £ a, or in the X-plane, X6 a =^-X, X , -X € a, yield the following rather generic spectral profile. Figures 5,5 (Compare Figures 2,2 .) / X-plane Figure 5 Generic Profile of the Floquet Spectrum a(X) 26 mma Hkww '/MM VJM E-plane Figure 5^ Generic Profile of the Floquet Spectrum a(E) As the period L becomes infinite, the spines go away and the entire positive real E-axis becomes continuous spectrum (radiation in the sine-Gordon field), the conjugate pair of bands in the E-plane shrinks to conjugate poles, a breather, and the three bands on the negative real E-axis shrint to poles and describe three kinks (or antikinks). (Refer to Theorem 1.III.2, part (iv).) Thus, Figures 5,5'' collapse to Figures 2,2^ in the infinite-period limit. Based on these whole-line analogies for the limiting configurations, we see the conjugate pair of bands must be related to trains of breathers while the bands on the negative real E-axis correspond to trains of kinks or antikinks. 27 l.IV Finite Degrees of Freedom l.IV.l 2N Invariant Simple Spectra {E.}; Determination of a Riemann Surface. Material in previous sections Indicates that the elementary excitations of the periodic sine-Gordon field consist in trains of kinks, trains of breathers, and trains of radiation. (Of course, the distinctions may not be as well-defined as in the whole-line case.) For the remainder of the paper, we seek qualitative and quantitative connections between the spectrim a and the physical characteristics of these basic excitations, such as frequencies, dispersion relations, and amplitudes. First, we use the invariant spectrum a to fix a Riemann surface which is used to integrate the periodic sine-Gordon equation. In general, we seek all solutions of the sine-Gordon equation for which the spectrum a is prescribed. Formulas such as p(E) = "IIaCe) + /a^(e) -4) for the Floquet multiplier p(E) indicate that such potentials will be fundamentally related to the analyticity structure of the function /A^CE) - 4 = /(A(E) -2)(A(E) +2) , 28 which has branch points at E = 0,",® and the simple periodic and antiperiodic eigenvalues spectrum a consists of bands that terminate precisely at the branch points of A^-4 , so that the branch cut structure of /a^(E)-4 can be chosen to coincide with the spectrum oJ The appropriate function theory of the sine-Gordon equation should derive from the two-sheeted Riemann surface of (E)-4 : two copies of the E-plane with branch points {Ej} (including 0,") consistent with the constraints of Theorem l.II.l, "k Ej a branch point => E^ a branch point. (l.IV.l) Since there are, in general, infinitely many simple periodic and antiperiodic eigenvalues, A^(E)-4 will have infinitely many branch points, leading to a Riemann surface of infinite genus. ® The branch points at E = 0, arise not as zeros of A^-4, but due to the essential singularities of A(E) at E = 0,<»; see Appendix F.l for explicit details. ^ This is completely analogous to whole-line scattering theory: The (fi?) Green's function G is an analytic function of the energy parameter E except for essential singularities at E = 0," and poles at the bound state energy levels; G(E) has a branch cut which can be chosen along the positive real axis to coincide with the continuous spectrum. For the periodic problem (1.II.2), the whole-line Green's function is an analytic function of E, except for essential singularities at E = 0," and branch points at the periodic and antiperiodic eigenvalues; the branch cut structure can be chosen to coincide with the spectrum a. However, the special case of exactly (2N) simple zeros of A^CE)-4 provides sine-Gordon fields which contain a finite number (N) of basic excitations. In that case (see Appendix F.l) p E /2N+1 (E)-4 = C n (.1 - n Cl-:^) / n (E-E,); (l.IV.2) j>2N+l j j<0 Vk=l the number of branch points is finite, leading to a finite genus Riemann surface. This case of N degrees of freedom is potentially useful in applications. For example, consider a study of the sine- Gordon equation with arbitrary periodic initial data. First, spectral analyze the data to determine the structure location of the spectrum 0. Next, retain only the "most dominant" bands to obtain a finite-band approximation of a. This finite-band spectrum then yields an approx imation to the full wave in terms of a finite number of basic sine- Gordon excitations. With this motivation, we pose the inverse problem; the strategy is to prescribe (2N+1) branch points, {Ej^,k = 1,...,2N+1}, consistent with the symmetry stated above, and then construct the most general solution of the sine-Gordon equation which yields the spectrum (1.IV.2). (Refer to Matveev [30] and Dat6 [8 ].) Fix {E^,... such that E2jj^2^ = 0; for k=l 2N, Ej^ ^ [0,"), and Ej^ are either real (negative) or occur in conjugate pairs. Consider the Riemann surface of the function 2 2N+1 R (E) = n (E-E,). k=l ^ (For the case N = 3, this Riemann surface may be realized from the cut structure depicted in Figure 6. Notice the branch cuts 30 need not, in fact Matveev's [30] do not, coincide with the spectrum a; however, they could be so chosen.) The general sine-Gordon solution with this prescribed spectrum is an "N-phase wave train". These wave trains can be represented in terms of N variables, , each of which moves on the Riemann surface just specified. E '3 "4 oO E-Plane Figure 6 Sample Cut Structure for Genus N = 3 Riemann Surface 1.IV.2 y-Representation of the N-Phase Wave Train We now construct a representation of the N-phase wave train. We will follow an approach of Datfe [8] which is very straightforward and far more simple than alternatives in the literature. First we state the representation and then detail its construction. Theorem l.IV.l (ti-Representation of the N-phase wave train) With the simple spectrum fixed as J{Ej^,k=l,...,2N} , the general N-phase sine-Gordon wave train u(x,t) admits the represen tation N vN C-1) vi^Cx.t) u(x,t) = i log (1.IV.3) 31 2N where P =? 0, and the y.(x,t) satisfy ^ ^k' ^2N+1 k=l ^2 2N+1 , - (-1)" " ^ 11 + n M, n (u,-E ) 16P'2 j=l J j=l ^ ^ i ' 21 (1.IV.4) N n (vo-^4) j=i ^ ^ We refer to this representation as the "y-Representation of uCxjt)"; it constructs the general N-phase sine-Gordon wave from the /n\ simple spectrum J . We complete this section with the very simple proof, due to Date , of this Important theorem. The linear Takhatajian-Faddeev space-time systems, which carry the sine-Gordon equation as a compatibility condition, are T given by equations (l.II.3a,b): C^p = ) 0 -1' '0 1' e 0n X ^ + T W ^="5. (1.IV.5) X il 0 / 16/e \0 e~^" Date employs a completely equivalent linear system for quadratic eigenfunctions {f,g,h} , where » 8 = ^^ > = ' (l.IV.6a) The vector function 32 ? = Cf,g,h)"^. Cl.IV.6b) satisfies the following system which is easily derived from (1.IV.5); 0 16v^ I ^ 16v^' -iu) c F = 2i /e + -•jw F. (1.IV.7) X 16/E iu e 2i + |w 16/E (This system also carries the sine-Gordon eqtiation as a compatability condition, F = F .; XU use Date then seeks solutions of (1.IV.7) which are polynomial in the eigenvalue parameter , 1 N I N N . f = — I f.E^ . g= I g.EJ ,h=^ h E^, (1.IV.8) ''E j=l J j=0 ^ j=0 ^ and shows this ansatz demands that the potential UjW are N-phase waves (note the polynomial degree N will label the number of phases in the sine-Gordon wave). Moreover, a short calculation, which we now present, provides an exact representation of these N-phase waves. That is, if we insert the polynomial ansatz Cl.IV.8) into the •iiff®^£iitial equations (1.IV.7), the following system of equations for the coefficients {f., g., h,} results : J 3 J 33 T. i _iu. 16 ® •*• ^^j-1 ^ 16 ® "^j' ^ 2if^ + g e - 2 wgj , j=0 N CI.IV.9a) t (h.) 21f. +1 fj+i+fwh. » j=0 N 3 X t subject to the constraints f = f MJ.1 = 0 o N+1 (l.IV.9b) iu , -iu, e g + e n = 0. The last constraint is crucial; it gives a formula for the potential u(x,t) in terms of the coefficients fact, minor manipulations with (l.IV.9a,b) lead to the important results u(x,t) = "I log (1.IV.10a) N w(x,t) = 4 (1.IV.10b) 'N plus two additional facts. % ^ " constant. (1.IV.11a) h g = constant, (1.IV.lib) o o We are now close to the main goal of constructing the N-phase potentials u,w; as soon as the polynomial ansatz is justified and 34 the structure of the coefficients g^,h^,fj^ is found, we will have a representation for the potentials. We conment now that substitution of the relations (l,IV.10a,b) for u,w into the system (1.IV.9) yields a closed system of differential equations for the coefficients f^jg^jh^; the existence of polynomial solutions is thereby guaranteed by the solvability of that system. (The converse is also true.) Thus, the polynomial ansatz is justified. But more importantly, the constraints involving the potentials u,w very quickly lead to the N-phase wave solutions of the sine-Gordon equation. From the system (1.IV.7), it follows that „ 2N . 2N P(E) = f - gh = I P.E^ = n (E-E.) (1.IV.12) j=0 ^ j=l ^ is a polynomial with constant coefficients (first show the function (f -gh) has vanishing x,t derivatives by (1.IV.7), then use (1.IV.8)). Now let Uj(x,t), j=l,...,N, be the roots of g = 0; then N g(x,t) = - n (E-y.(x,t)), (1.IV.13a) j=l ^ with g = (-1) n y.(x,t). CI,IV.13b) j=l ^ 35 Since u(x,t) = ilog , we seek h . But ' o 2N P(E=0) = f-gh =-gh s= n E,, and we find o o "o o j' g. 2N n E. j=l ^ Therefore, the sine-Gordon solution u(x,t) has the representation N vN (-1) ^|j^u^(x,t) u(x,t) = ilog (1.IV.14) 2N n E. This representation clearly illustrates N degrees of freedom in the sine-Gordon wave; to be useful, we must understand the behavior of the variables Thus, the next step is to derive differential equations for the Pj(x,t). Differentiating g(x,t) in (1.IV.13a) yields N N = I ^ (E-u (x,t)). (1.IV.15) ^ j=l ^ ^ k=l ^ But g(x,t) also satisfies the system Cl.IV.7), so that -xu (g) = 2i(v^ + ) f - "I wg ^ 16>^ ^ 36 now evaluate this equation at E = y^(x,t) and use (1.IV.15) to find N (1.IV.16) k=l 16^' Then is found from (1.IV.12) and e is given in (1.IV.10a); , 2N f (y.) = n (Vn-E.), ^ j=l ^ J N vN -iu '2N n E. j=l ^ Inserting these into (1.IV.16), we arrive at the "y-Representation of u(x,t)": N .N (-1) y;^(x,t) u(x,t) = ilog (1.IV.17) 2N n E. j=l ^ where the satisfy the system of differential equations .N N 2N+1 V2 2i I 1 + (1.IV.18) tr72 3=1-n. jli 16P N n (yn-pJ j=i ^ ^ 37 In summary, we note this ''y-Representation" parameterizes the N phases by the variables which lives on the Riemann surface of R 2(E) = 11 (E-E.); the behavior of the j=l ^ variables can be deduced from the system of differential equations (1.IV.18). Moreover, we emphasize the ease with which the Dat^ approach recovers the potential u(,x,t) from the prescribed simple spectrum = {Ej^, k=l,...,2N}, 1.IV.3 0-function Representation of u(x,t); N-Phase Wave Train Although we feel the "y-Representation of u(x,t)" is the most useful for studying the behavior of the N degree of freedom sine-Gordon wave, it turns out that the y-equations (1.IV.18) can be integrated exactly in terms of 6-functions to yield the "0-function representation of u(x,t)". The main advantage of the 6-function representation is that it replaces the N degrees of freedom y^,...,yjj by N phases, ii^(x,t),... ,J!,jj(x,t), each of which is linear in space (x) and time (t). However, since this representation is not central to our discussion, we refer the reader to reference [30] for the derivation and merely give the representation. To state the results, we first describe appropriate cycles and differentials on the underlying Riemann surface which are used to define the N-dimensional 6-function. 38 2 2N+1 On the Riemann surface of R (E) =? II (E-Ej^), ^2N+1 ^ ^ k=l (Section l.IV.l) we introduce two families of closed curves (which form a basis for contour integration and are typically referred to as "canonical cycles"), {a^,a2,. ^ and {bj^,b2, • •.,bjj}. We illustrate these a-b cycles for the special case N = 3; from the cut structure of Figure 6, the precise paths are depicted below in Figure 7. a-cycles for N = 3 upper sheet - - -4 lower sheet \ b-cycles for N = 3 Figure 7 a,b-cycles for Genus N=3 39 Next, introduce N differentials (Abelian differentials of the first kind), N-1+. ..+c,„ dU^ = — dE, v=l,2 N , (1.IV.19a) where the matrix of constants C = is fixed by the normalization conditions <> dU = 6 . (1.IV.19b) ^ V vy From these differentials, define the period matrix B = (B ) by VV' B = A dU, . (1.IV.19c) pv J V b y With these ingredients, we construct the N-dimensional 9-function 0(?;B) = ^ exp{Tri(B^,^) + Zirl (We remark that as defined, Im B is positive definite, which yields very rapid convergence for these series - another advantage of the representation to follow.) The y-equations (1.IV.18) are then integrated using 0C?;B) to give: 0 (t(x,t) + Y ; B) u(x,t) = 2iS,n (1.IV.20) 0(t(x,t);B) 40 where = -2i([C^,-H^ C^^]x + t) + ^.^(0,0) 2N ^2N+i = o» P = \ ' t + -| = Ui + f. ^2 i ^N • Here, the constant 1^(0,0) must be chosen to guarantee a real solution. We refer to equation Cl«lV.21) as the "6-function representation of u(x,t)"; notice that it explicitly displays the sine-Gordon solution u(x,t) as a multiphase wave train. That is, it has N phases, {£^(x,t), v=l,...,N}, each of which depends linearly on x and t. The wave form is parameterized by the fixed /n\ simple spectrum ^ = {E^^,... ,E2j;j} and the N constants Jl^(0,0) Jijj(0,0). The branch points £^,...,£2^^ characterize the wave by fixing its wave numbers, frequencies, amplitudes, periods; the constants ]t(0,0) center each phase. The solution u(x,t) will be of little use until the connection between these input parameters, {E^,...,£2^^}, and these physical characteristics is clearly understood. In the next section, we discuss this connection in great detail for the N = 1 case; in the final section, we consider a special class of N = 2-phase waves. 41 l.V Traveling-Wave Solutions (Single-Phase Wave Trains) l.V.l Reduction to N = 1 In this section we fix the number of phase N = 1: eacx,t) +|; B)] u(x,t) = 2iJ!,n ea(x,t) ; B) I ' (l.V.la) where the single phase S,(x,t) is given by, with P = £^^£2 A • - 1 J!,(x,t) = -2iC[(l + i7')x + (1 - i7-)t] + S,(0,0). (l.V.lb) ,16P- ^ 16P ^2 From this representation we see that the N = 1 sine-Gordon solution is a traveling wave; that is, a function of x and t only through the one linear combination (K.X + cot), with "phase velocity" U given by 16P^2_I U = ii = j; . (l.V.lc) 16P^2+2. l.V.2 Direct Ansatz Method to Display Traveling Waves Once we realize the N = 1 solution u(x,t) is a traveling wave, there is a direct and elementary approach which catalogues all single-phase traveling-wave solutions of the sine-Gordon equation. One merely seeks a solution of n ^ - u + sin u = 0 in the form tt XX of a traveling wave: u(.x,t) = U^(KX + OJt). 42 Inserting this ansatz into the sine^Gordon equation yields an o.d,e. for UJ as a function of the "phase" 6 - KX + ojt, 2 2'''' (u) -K )u^ = -sin u,j. , where (•) (')• This "effective oscillator" equation may be integrated once to obtain the energy equations (u^)^+ = E, if = ^ < 1, (l.V.2a) + Veff = £ = -E, if = ^ > 1, (l.V.2b) where ^gff ^'eff^'^T^ effective potentials given by ^eff^^T^ = "T ^eff^^T^ = The potential energy diagrams are sketched in Figure 8. As we catalog all traveling waves, we first classify waves as having phase speed |u| = which (i) exceeds, or (ii) is exceeded by, the characteristic speed c = 1, For phase velocities satisfying U 2 < 1, there are three classes of solutions which are depicted in Figures 8a,b,c. Notice when the energy parameter E = 1, the traveling wave is a single kink which rises steadily from -2IT to 0 (this corresponds to picking the positive square root when solving for u^ from (l.V.2a); the negative determination yields a single antikink which falls steadily from 0 to -2Tr). For E > 1, u^ is a monotonic sequence of kinks (antikinks), and should be thought of as a "kink train". (We justify this interpretation in Section 1.V.4). U^<1 <=> > 0 -TT Kink Train -rr ITT Kink (Soliton) -n-Uo -rr Xo-L Pure Oscillatory State Figure 8 Potential Energy Diagrams and Corresponding Solutions (N =1) U^>1 <=> Kink Train Kink (Soliton) -TT Pure Oscillatory State Figure 8 Continued 45 This kink train is periodic in 9 (mod Zir) with period P as computed directly from the energy equation (l.V,2a); Ztr du,^ ^=5 P(k,U),E), E > 1. (1.V.3) /E^cos "T When -1 < E < 1, the traveling wave becomes oscillatory about u^ = -IT. The solutions are strictly periodic for this range of E with 6-period 2P, -ir+u (E) , o du,_ ^ = 2PfK.ai.E'). - 1< E< 1. (1.V.4) -.-u (E)Ve-cosUT o In this formula, the "turning points", -ir + u^(E), denote consecutive zeros of E - = E -cosu^, as depicted in Figure 8c; these turning points fix the amplitude of oscillation by Amplitude of Oscillation = (1.V.5) (Notice that the spatial period L and temporal period T can also be computed from the energy equation (l.V.2a) simply by changing variables from 6 to x and t, respectively. From these, the 27r Ztt wave number k = — and frequency lo = immediately follow.) 2 When U >1, the phase speed exceeds the characteristic speed c = 1, and the Figures 8d,e,f apply; in this case the effective potential is = -cosu^ and the total energy is £ = -E. When € = 1, is a kink which rises from -IT to ir (this is for the 46 choice = + /( ^ in the energy equation Cl.V.2b), the other choice yields on antikink which falls from TI to -TT). For ^ > 1, a kink train results; for -1 < S < 1, the traveling wave u^ becomes a truly oscillatory state about u^ = 0. In all cases the periods may be computed (as above) from the energy equation Cl«V.2b). The main point of this subsection is that, using this "effective oscillator" approach, one can quickly catalog all traveling waves as belonging to one of six distinct classes< Once the phase velocity U is fixed by either |u| (U = E) admit clear physical interpretation; they characterize the wave number, frequency, and amplitude of the wave; the phase velocity is given by ^ and the 6-period, x-period, t-period are all defined by loop integrals. The y-Representation of u(x,t) for N = 1-phase waves has input parameters E^, E2. The location of E^, E2 in the /q\ ^ Q ^ complex plane (subject to the constraints E^ ^ I ^ ^ and Ej ^ [0,<»)) must yield the same catalog for the traveling-wave solutions. We need a feeling for this catalog and the way gauge the physical cha,racteri.stics of the sine-Gordon wave. We discuss these topics next. 47 1.V.3 ]i-Representation of u(x,t) for N = 1; Catalog and Physical Characteristics of the Traveling Waves In Terms of = {£^,£2} /q\ For the N = 1 case, the simple spectrum ^ contains and E2. From the symmetry E^ € =>Ej € (Theorem 1,11.1), together with the fact (Section 1,111) that E > 0 is all continuous spectrum (with no gaps), we find that the branch points ^^^,£2 are located in the complex E-plane in only two distinct ways: Case 1. Ej^ < E2 < 0, so both lie on the negative real E-axis. Case 2. E2 = E*, E^ E2J so they occur as a conjugate pair. These possible locations of E^, E2 lead to the three configurations depicted in Figure 9, which also displays the canonical branch cuts. 4 El E2 E„=E Case (i) Case (ii) Case (iii) Figure 9 Various Locations of with Appropriate Branch Cuts (N = 1) 48 We now use the analogy with whole-line sine-Gordon scattering theory C refer to Theorem 1.I1I.2 and Figures 2^,5^) to guess the excitation classification based on the location of and ^2' Case 1, < E2 < 0, appears to arise from an isolated pole ^ which spreads along the negative real axis into a band of spectrum; this should be the periodic analog of one kink, namely, a train of kinks. ^ A Case 2. E2 = E^, E^^ 5^ E2, can be viewed with E^^, E^^ located near the positive real axis, which is all continuous spectrum related to radiation degrees of freedom. It then seems that the band of * spectrum connecting E^, E^^ should give a pure oscillatory excitation, known as "plasma" or "Josephson" radiation. We will verify these guesses next using the N = 1 VI-representation of u(x,t); in the rest of the chapter we contrast the physical characteristics within each class of solution (that is, within each cut structure) based on the relative location of E^ and E^. Recall the N = 1 y-representation of u(x,t): '-ji(x,t) \ u(x,t) = ilog I n , (l,V.6a) where p(x,t) satisfies (p)^ = 2i 1 + /y(y-E^)(y-E2) . (l.V.6b) 49 We aim to deduce many facts directly from (l.V.6a,b), without need of the exact integration of the (li) equations in terms of t 0-functions. First, the relation between u(x,t) and vi(x,t), (l.V.6a), together with the (1^)3^ equations, quickly yields t u y 16(E,E ^ _t^_t 12 ^ X ^x 16(£^£2)^ + 1 Thus, the ^-representation clearly shows the N = 1 sine-Gordon solution is a traveling wave; that is, U(x,t) = Xl^CKX + Ut) , with the phase velocity U = ^ , from (1.V.7), given by 166 (£^£2)'^( - U = ^ . (1.V.8) 16(£^^£2)+ 1 (Note that this is the same result found from the N = 1 0-function representation. Section l.V.l.) We therefore find the phase velocity is a function of £^ lA and £2 only through the product (£^^£2)' . We sketch U as a 1/ function of (£^£2) in Figure 10; note the following facts (c = 1 is the characteristic speed): (i) (£^£2)< 0 <=> |u| ^ 1; the sign (+) determination of (E2^£2) fixes the phase speed relative to the characteristic speed. u .h Figure 10 Graph of U vs. 51 (±i) |(Ej^E2)^^| > <='> U > 0 ; the wave is at rest when 1(E^E2)^1 lies on the circle of radius in the E-plane and travels to the right (left) when I(£^£2) I lies inside (outside) the circle. Moreover, now that the y-representation has implied traveling waves, we can make contact between the y-representation and the "effective oscillator" approach: The first step is provided by the formula (1.V.18) for U in terms of £^,£2; we now need to relate the energy parameter E to E^ and £2* To do so, recall the energy equation (l.V.2a), •|(K2-to2)(u^)^ + cos u^ = E 3 d which can be written in terms of x (-^ = K^) •|(U^-1)(u^)^ - cos u^ = - E. (l.V.9a) Now use the y-representation to gain another representation of (l.V.9a). 2 4(1+16(ETE„)'2) (y-E^)(y-E2) 2 u X ~ 2 Note from (l.V.8) that ydJ^-l) - thus V 2 ' (H-16(E^E2)'2) 52 -4C32) [(Ej^E2)'''2e^"+Ej (16CE^E2)^2) -(E^E2)^2e^'' = 2^ [cEiE2)Ce2^" + l) H. CE,E2)'''^ei"(E^+E2)] = COS u + *2 T. * CE^Ej)'^ / -E (l.V.9a) •^(U1^it2 -l)u IN 2 - cos u = ^ , ^2 ^ (l.V.9b) 2 ^ Therefore, for U < 1 <=> '= ^ total energy E satisfies Case 1. E^ < E2 < 0 (l.V.lOa) E > 1. Case 2: El - E2 , El ^ ^2 E = - cos (phEi) (l.V.lOb) -1 < E < 1. upper sheet lower sheet Figure 11 y-cycle; N= 1, < E2 < 0 54 2 Vo For the choice U > 1 <=> (£^^£2) ~ exact same formulas hold with the total energy E replaced with -E. We now see that the location of £^, £2 catalogs the traveling- wave solutions in the same way that the effective oscillator parameters (U,E) do. In summary. Case 1. E^ < E2 < 0 => "Helical wave" solutions, Figure 8a,d. Case 2. ~ ^2 ' ^1 ^2 Oscillatory solutions, Figure 8c,f. Moreover, in the limiting case E^ = £2 < 0 (the transition state between Case 1 and Case 2), the total energy E = 1 and the resulting wave form is a single kink - the sine-Gordon soliton. Thus, Case 3. E^ = E2 < 0 => "Soliton limit" - a single kink. Figure 8b,e. We now turn to the physical interpretations of the parameters Ej^ and ^2- Some of this information is already provided by the discussion surrounding Figure 10 for the phase velocity U. Next we show how the amplitude of the oscillatory states is fixed by the location of E^ = £2 , E^ 5^ £2" 2 Fix U < 1 (the other case follows in the same manner). There are several ways to argue the amplitude behavior based on the •k location of ~ ^2 ' ^1 ^ ^2' consider the y-representation for this cut structure. From u(x,t) = i log| j ^ reality \ '^1 of the solution implies ly(x,t)| = /£^E2 = [EjI: p(x,t) resides 55 on the circle of radius |Ej^| in the cut E-plane, (see Figure 11) with ph(y) = -(Tr-hiCx,t)) , vU.t) - |Eje-U'+«Cx,t)) _ Since u(x,t) oscillates between |u(x,t) + ^ » we find the path of uCxjt) as shown in Figure 12. eo Figure 12 p-cycle; N = l, E^ = E*, The y-path clearly shows |phy| < ph E^^; since |phy| = |7r + uCx,t)|, we find u^ = amplitude of oscillation = ph E^, (l.V.ll) 56 Alternatively, we can arrive at this result from the effective oscillator diagram (Figure 13). --I Figure 13 Pure Oscillator (|u|U < 1); Effective Oscillator Diagram The amplitude u^ is determined by the roots (centered about -ir) of E = cos u; since we have just found E = -cos(phE^) = cos (phE^-ir) , the relation (l.V.ll) follows immediately. We remark that the amplitude of the oscillatory states depends only on the phase of E^, independent of the amplitude |e^|. At this stage in the presentation, we have used the N = 1 y-representation to show the sine-Gordon solutions are traveling waves; the input parameters (E^,E2) catalog all the traveling waves and describe certain physical characteristics (phase velocity and amplitude). For the remainder of Section l.V, we use the N = 1 9-function representation to arrive at an exact decomposition formula 57 for each class of solutions Chelical waves and oscillatory states) which gives insight into the fundamental building blocks for each solution. We also give explicit formulas for the physical character istics parameterized by and E^, and then contrast within each class of solution. 1.V.4 Infinite-Product Representation of 0-Functions and Summation Representation of the Traveling Waves Toda [40] used an infinite-product representation of the theta-function to interpret the periodic traveling-wave solutions of the Korteweg-deVries equation and the Toda lattice as a sim of soliton shapes successively shifted by one period. Generically, if we let u^(?) denote the periodic traveling wave with period L, and Ug(5) denote a soliton shape centered at 5 = 0, Toda's formula is of the form GO u^(C) = I Ug(?+nL) . n=-oo This formula allows the beautiful interpretation of the traveling wave as a soliton on a ring. It seems one of the remarkable properties of these completely-integrable nonlinear equations that such a formula is exact. In this section, we adapt Toda's arguments to the sine-Gordon equation; we show the "helical wave" literally is a kink train, while the oscillatory state admits an interesting and unexpected interpretation. 58 We begin by recalling the well-known single-soliton solutions of the whole-line sine-Gordon equation. They are classified "kinks" or "antikinks", and are depicted in Figure 14, Kink Antikink Figure 14 Familiar Kink, Antikink Shapes The familiar formulas for these solitons, centered at x , are o u (x-x„,t) = 4 tan ^(e^''') , (1.V.12) Jx o AK (x-x^-vt) 1+ie''' u„ Cx-x ,t) = 2i Jin (1.V.14) Jx O AK where the choice of branch for An is taken to correspond with Figure 14. Now, the 6-function representation for the single-phase traveling waves is (equation Cl.V.l)) 0a(x,t) +j;B) u(x,t) = 2i Jin CI.V,15a) 0 (Jl(x,t) ; B) 59 where the phase Jl(x, t) takes the explicit form® (x, t) = -2i C (1 + ± •)Cx-x„) + (l-- •)t (l.V.lSb) le'cE^E^ 16CE^E2) With this representation, the wave number k and frequency oj are given by K = -4TrC[l + ] Im(B) 16(E^E2)¥ (1.V.16) 1 (*) = -4TrCri - 177] Im(B) 16(£^£2)'^ The choices (- "I" or + •^) are shown to yield kinks or antikinks, respectively. The normalization constant = C and period "matrix" B (see equations (1.IV.19)) are functions only of the eigenvalues E^, E2; explicit formulas for C and B in terms of elliptic integrals are given at the end of this section, but for now we note the general form: For E^ < E2 < 0 , B = ilm(B) , C < 0. (1.V.17) For E^ = E2 , E^ E2 , B =-•!+i Im(B), C < 0. ® For this single-phase case, Re[Jl(0,0)] is chosen to yield uCx,t) real, while lm[£(0,0)] centers the wave train. 60 Although the one-dimensional 9-function is defined by the infinite series CO e(£;B) = J] expCiirBn^ + i2'irJ!,n), n=-oo it also admits an infinite-product representation [ 44 ]. In Appendix B, we use these infinite products to show the 0-function representation (1.V.15) of the single-phase sine-Gordon wave has the series representation^ u(x,t) =l 2i&n/^"^^^ ]+ I 2iAn |(-1). j. (l.V.lSa) ll+ie ^ \ 1+ie ^ / where = K(X-X^) + a3t + 2mriB . (l.V.lSb) Now, by specializing to the two possible locations of E^, E2, using formulas (1.V.14) for the kink, antikink wave forms, and noting the facts (1.V.17), we find Theorem l.V.l (Decomposition Formulas for Single-Phase Sine-Gordon Waves) Fix Iu| <1 and the choice - ^ in £(x,t). Then (i) for < E2 < 0, 00 u(x-x^,t) = I {Uj, (x-x^-nL,t) + tr(.sgnCn)-l} (1.V.19) n=-co f- 1-ie = ^ {2iS,n + ir(.sgn(n)-l)}; (1,V.19'') n=i-oo il+ie^'^/ a ^ From these representations (1.V.18), a real implies ,1+ieJi, ^1="n 1 and the wave is therefore real. l±ie 61 Cil) for = E* , E2 , u(x-x ,t) = '5! {u (x-x -2nL,t) + u (x-x -(2n-l)L,t) o K AK ^ n=-oo + 2Tr(sgn(n)-l)} (l.V,20) - . ®2n\ / ^ . ®2n-l\ = I {2xAn( + 2i&n M^ , _ 2n \ . . °2n-l l+±e I \ 1-ie + 2Tr(sgnCn)-l)}, a-V.20'') where 0 = K(X-X -nL) + (ut, n o Zir KjOJjL = — are given by equations (1.V.16), and +1 for n > 0 sgn(n) = -1 for n < 0 We display these decomposition formulas in Figures 15,16. We now interpret these results: For Ej^ < E2 < 0, the helical wave solution, we find the wave is literally a kink train; a sum of translated kink shapes successively shifted by one period L. (We note the choice of (+ •^) in ACxjt) yields an antikink train.) Moreover, the wave train is explicitly periodic (mod 2ir), with spatial period L = MSL . (1.V.21) 2|C|(1+ ± 1^) 16(£^£2)^^ 62 frr •^L > Full Wave > Building Blocks Figure 15 Graphic Representation of the DecompoDecomposition s it ion Formula for the Kink ~Train, < E2E, < 0 2.i-, .3L cL Figure 16 Graphic Representation of the Decomposition Formula for the Oscillatory State, E^=E2» ^2 63 As and E2 coalesce, the period L becomes infinite Cuse the explicit formulas for B, C at the end of this section), the kinks (antikinks) in the sum move infinitely far apart, and only the n = 0 term remains; in this manner, the wave train reduces to a single kink (antikink) as E^^, ^2 coalesce along the negative real axis. lim L = + 00 J EJ-.E2 X-X -vt 1-16|E I o . V - , x-x -vtl lim u(x,t) = 4 tan ^ [ exp ° ' ; (1.V.22) =1^^2 /l-v^ As the branch points collide, E^ = E2 < 0, the branch cut collapses to a single pole on the negative real axis. We refer to this process as the soliton limit, since the resulting wave is a single kink (antikink) - the sine-Gordon soliton on the whole-line. This analyticity structure is depicted in Figure 17. From equation (1.IV.2), we note that a single pole of (A^(E) - •^) is analogous to a single pole of the transmission coefficient [a(X)] od Figure 17 Analyticity Structure for the Soliton Limit, E^^ = E2 < 0. 64 Thus, the collapse of the kink train to a single kink as E2 coalesce is consistent with our intuition from scattering theory and justifies the "guess" in Section 1.V.3 that the location El < E2 < 0 yields a kink train. Before this limit is taken, the series representation (l.V,19'') shows, for E^ < E2 < 0, that the sine-Gordon solution is a train of distorted kinks, or kink "shapes"; the speed and width each kink in the train is not that of the soliton except in the lone survivor of the infinite-period limit. The only effect of the interaction of the "tails" of the individual soliton components is to alter the speed and width from that of the kink in isolation to that of the wave train. For E^ = E2, E^ E2, the oscillatory wave solution, the series decomposition (l.V.20,20^) is even more interesting. It suggests that the fundamental building block of the oscillatory state is a kink- antikink pair, bound together to form the "bumps" in the wave. The full wave is then shown by this formula to be a train of kink- antikink pairs, successively shifted over each period 2L. The building blocks and full oscillatory state are depicted in Figure 16. Moreover, this representation shows the solution is truly periodic, with spatial period L given by (1.V.21). Once again, as E^, E* collide on the negative real axis, ~ branch cut collapses to a single pole on the negative real axis (Figure 18) . 65 E,=e; ©a Figure 18 Soliton Limit, < 0 The explicit formulas at the end of this section show that as E^, collide, the period L becomes infinite, and all kink- antikink pairs (except the, n = 0 one) move off to infinity. The n = 0 pair separates, the single kink centered at x = x^ survives while the antikink moves off to infinity; the resulting wave form is the single soliton (kink), given by the exact same formula as before, (l.V.22). Therefore, the soliton x-x -vt o u^(x,t) = 4 tan ^ exp /1-v^ I, 1-161E^I ^ ~ 1+161E^1 ' is the "transition state" between the two configurations E^ < E2 < 0 and E^ = E^, ^2' limiting wave form as either cut structure degenerates. Once again, before this limit is taken, the individual kink, antikink components are distorted kinks and antikinks, or soliton shapes. The interaction of the components alters the 66 speeds and widths from that of the solitons in isolation to that of the wave train; the correct speed-width relationship is recovered only for the single soliton that survives the infinite-period limit. Thus, these decomposition formulas provide insight into the nature of the single-phase periodic wave trains. We again remark, with amazement, that these representations are exact, and suggest a fundamental role for the whole-line soliton components in the periodic theory. We close this section with detailed formulas for the constants C and B in the 0-function representation, in terms of which the wave number k, frequency oi, and other physical characteristics are expressed. From the dependence of C, B on E^, E2, we then'contrast the physical characteristics of the single- phase waves based on the location of E^, E2. The normalization constant C and period "matrix" B are both expressed in terms of two fundamental loop integrals, which we denote 1(a), 1(b). With the branch cuts of Figure 9, we denote canonical paths by "a" and "b" cycles as shown in Figure 19, where we also depict useful contours of integration for each structure. * J-p O In terms of the holomorphic differential dl E , R (E)=E(E-E^)(E-E2)1 we define X(.a) =1 dl , Kb) = o dl. a-cycle b-cycle Then the constants C and B are explicitly given by = ifer' ® We now display the detailed information in Tables 1-4. 67 <^1 fit I oO Case (1) Ei oc> Case (ii) E2 = E*, I" < phE^ *m^i^ 3e Case (iii) Useful Contours of Integration Canonical a,b cycles ^2 = E*, 0 < phE^ Figure 19 Canonical a,b-cycles and Useful Contours of Integration for All N = 1 Cut Structures 68 Table 1 General Structure of the Loop Integrals 1(a), 1(b), Normalization Constant C, and Period Matrix B. In the notation of Figure 19, 1(a), Kb) have the following representations in terms of the contours a^, the explicit facts then follow by routine complex integration, with C = » ® " Ka) • Case (i); I(a) = 2 dl ; Kb) = dl "I '1 -^1e. =>I(a) < 0. => Rel(b)= 0, Iml(b)< 0. => C < 0 , B = i Im B, Im B > 0. Case (ii): 1(a) = 4 Re dl ; Kb)=-yKa)+2 dl + 2i Im dl =>Ka) < 0. => Re Kb) = -j Ka), Im Kb) < 0. => C<0; B = --2 +iIiiiB, ImB>0. Case (iii): 1(a) =4 dl + 4 Re dl ; 1(b) =- jKa) + 2i Im dl a, =>Ka) < 0. => ReKb) = - ^Ka), ImKb)<0. C<0; B=--|+iImB, ImB>0. 69 Table 2 . Elliptic Integral Formulas for the Normalization Constant C and Period Matrix B The loop integrals 1(a), 1(b), and thus the normalization constant C and period matrix B can be expressed in terms of familiar elliptic integrals [5 ,44]. In detail, (refer to Figure 19 and Table 1). Case Ci)5 E^ < E2 < 0 " «(s) „ _ IK^(s) ° " K(s) where the modulus s 2 = -^1 Case (ii): E^ = E* , E^ 7^ E2 , f lphEj^< ir C = 4K(s) B = - 1+2^ [F( where s^ = •|-(1 + cos(ph E^)) , s^^ = •|-(1 - cos(phE^)) (j) = Cos ^(rTr) > ^ = Sin ^C^) . s Case Ciii): E^ = E* , E^ E2 , 0 < phE^^J C = 4[F((|),s) + F(i/J,s)] B = _ ii iK(s^) 2 2[F( where s,s^, Table 3 Soliton Limit = E2 < 0) Formulas From the explicit elliptic integral formulas in Table 2, the "soliton limit" as , E^ collide on the negative real axis is computed. The results are the same for each cut structure. El < E2 < 0 and E^ = E^ » E^ 5^ E2. In detail (refer to Figures 17.18). dE lim Kb) = 2 = -X" (E-E^).^ Ei^E2 f dE _2TI- lim 1(a) = (E-E,),^ = [Residue at E^] = — 1^^2 lE-E^h € ^ lim L = + <*> Ei^Ei 1 1-161E^I lim K(x-x ) + a,t = ^ [x-x^-vt], v = i+igjE | (for the choice |u| < 1) 71 Table 4 Contrasts in Physical Characteristics of the Oscillatory States in Terms of . To display the difference in the (sub-characteristic speed, 9 ^ U < 1) oscillatory states based on the relative location of El ^ we consider two limits that essentially "cover" the E-plane. First, we display the "angular dependence" of the physical character istics, and second the "radial dependence". 1. Fix IE^Inconstant, and consider ph E^ ; u -»• 0 (See Figure 20) E = Energy: + 1 -1 . u^ = Amplitude of Oscillation = phE^: IT -»• 0 . 1 K = Wave Number = ^]^(i+ : (—^ + -_) Q. 1 ->^T^ /. ^ 0) = Frequency = Airld (1- /p-r:;-): ( r— + ) -»• 0. I6/E1E2 4 Im B Separation distance between crests: + <» ^ 0. |c|oc: Width of each kink(antikink) component: /IE^I -V 0. We note that each of the above limits is monotone decreasing. 72 Table 4, Continued 2. Fix phE^= constant, and consider |e^|:0 <» E = Energy = -cos(phE^) remains constant, independent of |E^|. . = Amplitude of Oscillation = ph E^^ remains constant, independent [E^I . K = Wave Number; +<*> decreasing to =2 increasing to 4 (0= Frequency: + <» decreasing to -<*> , *"^1 |£ [ - 16 ~ ^ ' ImBa Separation distance between crests remains constant. lc| a Width of each kink (antikink) component: 0 -»• + ». |E,I |E^| constant phEj^; ir 0 Figure 20 Graphic Limit as ph E^ : IE^I constant 73 l.VI Separable Solutions (N = 2) l.VI.l Definitions and Motivation Separable solutions of the sine-Gordon equation, u.^ - u + sin u = 0 , (l.VI.l) tt XX are defined by the ansatz that the x,t dependence separates, for example, u(x,t) = 4 tan ^(f(x)g(t)) , (l.VI.2) which was initially suggested by Lamb [26] in the context of optical pulse propagation. Since then, Costabile et al. [7 ] and Fulton [17] have considered these special solutions in their study of the oscillatory behavior of a one-dimensional Josephson transmission line (JTL). The most detailed mathematical study of the "separability" of the sine-Gordon equation is given by Osborne and Stuart [35]. The salient feature of- the above references is the fact that the separable ansatz, (l.VI.2), leads to elliptic function solutions for f(x) and g(t); moreover, from references [7 ,17] it is clear on physical grounds that these oscillatory sine-Gordon solutions have at least two degrees of freedom, and therefore represent the nonlinear interaction of two or more periodic traveling waves. However, we know (refer to Section l.IV) that the general function theory for periodic N = 2-phase waves leads to hyperelliptic functions on the genus 2 Riemann surface of R^(E) = E n (E-E,), = {E ....,E,} , k=l 74 not to elliptic functions on a genus 1 Riemann surface. The question naturally arises as to the source of this degeneracy in the function theory. In this section, we identify and describe this degeneracy. We will analyze one special case, defined in [ 7] by the "open circuit" boundary conditions, u (0,t) = u^(L,t) = 0 ; (1.VI.3) X A the authors give explicit 2 degree of freedom separable solutions satisfying these boundary conditions. There are many other types (as discussed in [35]) of separable solutions, which can be analyzed similarly. Our approach is outlined below. We first show that the "open circuit" boundary conditions, (1.VI.3), result from spatial symmetry and periodicity of the initial o ° data u(x), n(.x); therefore, these solutions fit into the general framework of 2-phase sine-Gordon solutions (Section l.IV), but with one additional constraint: initial data sirmmetric about x^ = 0. We then show how this spatial sjnmnetry implies the spectral symmetry: E. => —^— . With this symmetry in the simple spectrum ^ 16 E. J / Q \ J , the N = 2 y-representation is then used to show these special solutions are standing waves whose x,t flows separate and can be explicitly integrated in terms of elliptic functions. 75 1.VI.2 Even Initial Data (u(x), II(x)) <=> "Open Circuit" Boundary Conditions We first show the open circuit boundary conditions (1.VI.3) result from periodic initial data which are even functions about = 0. The converse is much easier and does not require proof. We state the result in Theorem l.VI.l Let u(x,t) be a solution of the sine-Gordon equation, (l.VI.l), with smooth, periodic initial data, u(x,t=0) = u(x), 0 u^(x,t=0) = II(x), even about x^ = 0 : u(x+L) = u(x) (mod 2ir) u(-x) = u(x) 0 0 (1.VI.4) n(x+L) = n(x) 0 0 ii(-x) = n(x) . Then: u(x,t) is even and periodic in x for all time t, and satisfies the "open circuit" boundary conditions at x = 0, ±L, ±2L,... . That is, u(x,t) satisfies (i) u(x+L,t) = u(x,t) (mod 2ir) (ii) u(-x,t) = u(x,t) (iii) u^(.0,t) = u^(nL,t) = 0. Proof of Theorem: Part (i) is well-known; see, for example, McKean [ 31 ] . To prove Part (ii), we begin with u(x,t) satisfying the 76 hypotheses (u(-x), n(-x)) = (u(x), II(x)), Now define u^(x,t) by u^(x,t) = "ICuCx.t) ± u(-x,t)) . We aim to show u (x,t) = 0. It follows from the sine-Gordon equation that u^(x,t) satisfy the system (3^^ - 9 )u, + sin u, cos u = 0 ^ tt XX + + - (9^^ - 9 )u + sin u cos u, = 0 tt XX - - + u^ (x,0) = u^(x) 0 9^ u^ (x,0) = II^(x) . S(-x) S(x) u (x) = 0 But implies 0 0 n(-x) n(x) n (x) = 0 , so that u (x,t) = 0 , proving Part (ii). Part (iii) follows rather 0 ^ o ^ easily. Since u(x), n(-x)) = (u(x), n(x)) implies 9 u(x,t) = 0. Periodicity then gives u(x,t) = 0 for any * x=0 x=nL integer n. 1.VI.3 Spatial Symiuetry <=> Spectral Symmetry We now use the Floquet theory of Section l.II to characterize even potentials by a sjrmmetry in the spectrum. 77 We begin with Theorem 1.VI.2 Consider the Takhatajian-'Faddeev eigenvalue problem, (1.II.6), with periodic, even initial data (u(-x), n(-x)) = (u(x), II(x)) uCx+L) = u(x) + 2t7M, M = "charge" of u(x) 0 0 n(x+L),= n(x) . Then; the Floquet discriminant, A(E), satisfies A(-^) = (-1)^ A(E) . le^E The proof is given in Appendix C (we also note the converse is true); it is now an easy excerise to deduce symmetries /q\ in the simple spectrum . In particular, we find Corollary 1.VI.2 Under the assumptions of Theorem 1.VI.2. E. ^ => . (1.VI.5) 16 E. J Consider the case N = 1. From Section l.V, we know the N = 1 /Q\ simple spectrum = occurs in only two forms: N=1 = {E^ < E2 < 0} or = E2 , E^ 5^ E2}. For the kink train case, N=1 E, < E„ < 0, the symmetry (1.VI.5) implies E_ = —I— . ^ le'^Ej^ 78 But the phase velocity U satisfies CSection l.V) 16(E Ej'^-l U s =—= 16(E^E2)^1 since U cannot be infinite^®, we find that the only "open circuit" kink trains are at rest (U = 0). The oscillatory case. E^ = E^ > ^2. ^2 ' same result. Physically, this time-independent nature of "open circuit" single-phase solutions is quite obvious. In the JTL, u represents the magnetic flux (properly normalized). A single-phase traveling wave u will either transfer flux through the left (U < 0) or right (U > 0) boundary, violating the "open circuit" boundary conditions; thus, stationary waves. The physical cosiderations of [7 »17] imply non-stationary separable solutions must be standing waves. This forces N ^ 2; next, we investigate N = 2-phase solutions (Section l.IV) in the light of this spectral symmetry. 1.VI.4 Consequences of the Spectral Symmetry (N=2); Separability, Standing Waves, and Elliptic Functions We now use the discovered spectral symmetry, which characterizes the sine-Gordon solutions satisfying "open circuit" boundary conditions, to show first that these solutions are, in fact, separable, and second, to explicitly describe the degeneracy in the function theory. Our analysis is through the N = 2 ^-representation of u(x,t) (Section l.IV): fS") u = 0° => u(x,t) is independent of x => u(x) = constant => ^ is empty. 79 [y.(x,t)u9(x,t)\ u(x,t) = i log I q 1, (l.VI.6a) 4 where P = II E., e {E. E,}, and y. _ satisfy j=l ^ J. f j.,z (y.) =2i I 1+- i (l.VI.6b) X I 16?"^ I X? First note that the spectral symmetry E, 6 ^ le^E." J forces = {E-,E„, —i— , —I—} , (l.VI.7a) ^ 16^E^ 16^E2 2^1 R (E) = E n (E-E.)(E 5—) (l.VI.7b) j=l ^ 16 E. J 4 1 p = n E. = ^ . (1.VI.7C) j-1 ^ le' With J given by (l.VI.Va), there are only two possible configurations of the spectrum; for convenience, we list and graph these below, together with the branch cut structure and canonical a^-cycles. We note that the spectrum is invariant under the 1 X map E —~ , and the circle |e1 = -r^ is mapped onto itself by le'^E this transformation. This fact is manifested in Figure 21a,b; in each case there is a clear symmetry about the circle |E[ = ^ . (The entire analysis which follows rests on this observation,) Case 1 (Two Trains of Kinks, Antikinks) E, < E„ < —i— < — 1 2 ,,,2 16 E^ 16 Ej^ Case 2 (Breather Trains, Plasma Oscillations) E^.E*. -1- , , »ith |E I >1^ 16 E^ 16''E^ JL-. :k. oft (1,-o^cU. &i-(Jt^de a. Branch Cuts and a^-Cycles for 2-Phase Kink—Kink, Kink-Antikink, AK-AK Trains Under Open Circuit B.C. /•fei ufik " 1 •p/ o6 lb i k b. Branch Cuts and a^-Cycles for 2-Phase Breather or Plasma Oscillations Satisfying Open Circuit B.C. Figure 21 Branch Cut Structures for Two-Phase Solutions With Open Circuit Boundary Conditions 81 Before analyzing the y equations (1,VI,6b), we note one more consequence of the spectral symmetry. Define two holomorphic differentials on the Riemann surface of R(E), (l,VI,7b), dE dl R(E) (l.VI.8a,b) , r-EdE "- R(E) A simple but tedious calculation,involving no more than a change of variables, shows dI(E) = -16 dJ(€) -16 dJ(E) = dl(6), where 6 = —. 16 E Then, with the a^^-cycles as defined in Figures 21a,b, the following calculations hold for both cut structures; dl = 2 dI(E) ; a^^-cycle now change variables, E = , which yields 16^6 16^E„ I(aj^) = -2 " 16 dj(€) = 16 A dJ 1 •'a„-cycle 2 16 E, = 16 J(a2) Similarly, we compute 1(32) = 16 J(aj^) 82 We state this in Lemma l.VI.l For the cut structures of Figure 21, the a^-periods of the differentials dl = dE dJ = EdE R(E) ' R(E) are related by: I(aj^) = 16 J(a2) (l.VI.9a,b) I(a2) = 16 J(aj^) . With these facts, we turn to the y-equations (l.VI.6b), which h. 1 can now be written as (assume P = + —r-) le'^ 2i(l + 16y2)R(yj^) ("I'K - (l.VI.10a,b) 2i(l + 16UJ^)R(U2) (ivV or in the equivalent differential form. dy. dx R(y.) ^^l"^2^ ~ ^^^2^ \dt (l.VI.lla,b) dy. dx (y^-y2) = 2i(-l + 16y^).^^ 83 Algebraic manipulation then yeilds quite simple expressions for the flows: dy. dx = + 32i RCyj^) RCy2) dt (l.VI.12a,b) ley^dy^^ 16y2dy2 dx = 32i RCy^^) R(y2) dt We immediately deduce Fact 1; N = 2-phase solutions with the spectral symmetry displayed in (1.VI.7) are standing waves. To see this, successively integrate the -flows in (l.VI.12a,b) around the ^1*^2 noting that the variables ^^1*^2 dummy variables of integration. Then using the explicit relations (l.VI.9a,b) (for example. dy. • ISyj^dy^ 16y2dy2 + <1 ), I R(yj RCy^^) RCy2) a2 1 1 1 it follows that there are two phases in the wave train (as we know) with almost identical phases, except that the phase velocities are equal but opposite; standing waves. Moreover, simply adding and subtracting equations (l.VI.12a,b), we find 84 Fact 2: The -flows explicitly separate: (l+16iij^)d;i^ (l+16p2)dP2 + ;r7—V— = 64idx R(y^) R(p^) (l.VI.13a,b) (l-16ii )dii^ (1-16^2)^1^2 + :r7—c = -64idt. R(yj^) R(y2) We also conclude that the /^i-flow is characterized in terms of (n- one particular differential, Riemann surface of R(E), as opposed to the usual case of requiring the full holomorphic basis (see the 0-function representation. Section 1.IV). This suggests a transformation which exploits the symmetry displayed in R(E) and which reduces the -differentials to elliptic. The map that accomplishes this reduction is z=|-(E+^y-); (1.VI.14) ^ 16 E the resulting differentials for the -flows are: (14-16E)dE _ -16dz x-flow: ^2(z -•^) (z-z^) (z-z^) (l-16E)dE ^ -16dz t-flow: \/2(z 4-^)(z-z^) (.z-z^) where z = |(E +-Y-) . i = 1.2, ^ 16^E^ = {E , -j—, E , —5—} . ^ 16^E^ ^ 16^E2 85 Thus, we have found Fact 3: The integration of (l.VI.13a,b) shows that the -flow depends on an elliptic function which lives on the genus 1 Riemann surface of JCZ) = N/(Z (.Z-Zj)(z-z^) 1 1 We remark that the map z = y(E + —5—) is invariant to the spectral symmetry E. => —-— (which of course motivated its use) and ^ 16 E. •I therefore does not distinguish between the points E., — the two branch points become one in the image space, effecting the reduction to elliptic functions. The appearance of ±-^ as branch points for the j -flow is explained by two related facts, First, are the fixed points of the mapping z(E). Second, the OF symmetry in the E-plane, |E| = , gets mapped onto the 1 1 slit - Yg — ^ S Ye ' forcing the branch cut (Figure 22). _ L. lb Figure 22 Image of the Circle of S3nmnetry [E) = CHAPTER 2 CANONICAL VARIABLES FOR THE PERIODIC SINE-GORDON EQUATION AND A METHOD OF AVERAGING 2.1 Introduction In this chapter, we consider a modulational theory for N-phase solutions of a completely integrable Hamiltonian system. For these Hamiltonian systems, such as the periodic sine-Gordon equation, we use canonical action-angle variables to prescribe a Hamiltonian form of the modulational equations. We illustrate our approach with the N-phase sine-Gordon system as follows. First we give canonical variables for the full periodic sine-Gordon Hamiltonian system; next we reduce to N degrees of freedom (N-phase waves), and provide action-angle variables (^,^) for the 2N-dimensional phase space (however, the paths for the action integrals are not well understood yet). In terms of these ingredients, we then present our Hamiltonian prescription for the (2N) modulational equations. Finally we reduce to a single-phase sine-Gordon wave and deduce the important fact (see Chapter 3 and reference [13]): Riemann invariants for the single—phase modulational equations are provided by the N = 1 simple periodic, antiperiodic spectrum = {£^,£2} of the Takhatajian- Faddeev eigenvalue problem. This Hamiltonian structure of the modulational equations was suggested to us by Hayes [19]; however, our point of view is quite different. Hayes starts with a Lagrangian formulation of the reduced 86 87 system, then averages the Lagrangian a la Whitham [41], and finally performs a partial Hamiltonian transformation; we start with the full Hamiltonian canonical description, then reduce to N degrees of freedom, where we provide a Hamiltonian, indeed action-wangle, description of N-phase waves, and finally deduce the Hamiltonian structure of the modulational wave train directly from the underlying Hamiltonian structure of the N^phase waves. 88 2.II A Prescription for N-'phase Modulational Equations of Completely Integrable Hamiltonian Systems Consider a solution uCx,t) of the full Hamiltonian system. We assume u(x,t) is a solution of the full Hamiltonian system which appears locally an an exact, N-phase solution, u^(x,t), of the reduced, local Hamiltonian system. (These "full", "local" systems are defined below.) Thus, u ~ for such an approximation to remain valid over large distances in space and time, the parameters in the N-phase solution tijj(x,t) must vary slowly in x and t, thereby modulating the wave train (See Whitham [41] and Section 2.III). We seek a Hamiltonian description for the first-order modulational equations. Our arguments are entirely formal. Denote the complete Hamiltonian phase space by , where is an appropriate, fixed space of real-valued functions. The full HaTTi-t 1 tonian system is described by the Hamiltonian H, H: IR , 00 H(u) = / h(u(x))dx, u € (2.II.la) the Hamiltonian H then generates a flow given by (2.II.lb) 89 Here denotes some antisymmetric differential operator, (For the sine-Gordon equation, a vector formulation is required; see Section 2.V.) We now define a manifold of N-phase potentials /q\ Uj^(x,t) with fixed simple spectrum ^ = {E^,...,£2^^} • We have seen in Section I.IV that the class of N-phase waves which result from the Dat^ prescription in fact live on the Riemann surface of 2 R (E) = E n (E-E ), an N-torus; it is this manifold that we denote n=l by . By the explicit representations for Ujj(x,t) 6 given in Section I.IV (y- and S^function representations) we find that any function in is characterized in terms of by N phases, which are linear in space and time, thus by exactly N basic spatial frequencies, k = (K^,tC2> • • • > L H^(ujj) = lim /f h(Ujj(x))dx, u^^ 6 ^(k), (2.II.2a) L-^ -L and the generated flow is The manifold ZN-dimensional (with providing N constants of motion) and can be coordinatized by "action-angle" variables; (Jjj^j9^)» i = l,..,,N, (This statement is precise for sinh-Gordon [31] , presumably valid for sine-Gordon [31 ], and correct for the constant-mean KdV system Csee [14] and Chapter 4),) The complete integrability of (2,11,2) is manifested in terms of these action-angle variables by ^i = - ^ ' 6 = -r-^- = to. (the i temporal frequency); (2.II.3b) X i the angles 0^ also yield (0^)^ = (the i^^ spatial frequency). (2.II.3c) These wave numbers and frequencies co^ are fixed in the exact N-phase solutions u^^ of (2.II.2). However, we aim to approximate a solution u(x,t) of the full system (2.II.1) locally by u^^; by the opening remarks in this section, the parameters must vary on a large scale if this approximation is to remain valid. In order to distinguish the local and large scale behavior, we introduce two spatial scales, the local or "fast" scale, x, and a "slow" scale, X ; these are related by X = 0 (£ x), 0 < 6 << 1 . 91 Next we define a "subspace" »P of the full phase space , constructed by pasting together local phase spaces ^ indexed by the long scale X. The subspace ta is coordinatized by n (J(x), e(x,x)) (2.II.4a) xCfR where h ' (2.II.4b) 1^6= w(X) , (2.11.4c) and for fixed X, (J(X), "^(x.X)) 6 >^(K(X)) The idea now is to use the full Hamiltonian H, (2.II.la), but restricted to the subspace to characterize the dynamics of the slow modulations. Thus, we define a reduced Hamiltonian, H, H : x/ fR , H(u) = /" h(.u(^, X))dx, u € —00 Next we approximate the reduced Hamiltonian H by partitioning the X-axis into blocks of length AX; these blocks are moderate on the X scale, but long on the = -^ scale (Figure 23). Figure 23 Blocks on the X (long) Scale 92 i+j£ , , ,X H(u) = I h(u(|,X)) dX X=-oo X.l-ig 1, +00 == I X^)) dX (evaluation at midpoint) i=-00 X. u x-h +00 = y < h > AX. (definition of < h >) iJ =_oo i < h > dX , (Riemann sum) CO where X < h > ^•^h(u(|,xp) dX AX. X. u x-h _1_ fl^i-fi, h(u(y,xp) dy AX. T (-r) X^+L h(u(y,xp) dy L-x*" X^-L = Hj^(q(..X^)) . In this manner, we arrive at the approximate Hamiltonian, H n : J -y (R, H(u) = H^(u(-,X)) dX , (2.II.5a) and for fixed X, u(«,X) £ CK(X)) . 93 The reduced dynamical equations for the slow modulations in Ujj(.x,ty begin from the approximate action-angle formulas: • ca t-^ 3J t - 30 ->• 3H Since 6 is a fast variable, we differentiate e = — w.r.t. X, then use 6 ^ = K, equation (2.II.4b), to replace the top equation with -A 3H Now for the second equation, J = — , notice that H depends on 0 only through ~ (see equation (2.II.5a)); in this case, we find M ^ ^ = _e ^ 3? 30 3K A With these observations, the action-angle formulas lead to a Hamiltonian form of the modulational equations; 3t 3X [ J ' (2.II.5b) 9t " t- 3k(X) -I 94 Also notice that we may introduce a slow time scale, T = 6t, whereby the Hamiltonian structure (2.11.5) takes a beautiful canonical form: w (2.II.6a) aT ^9? / where is the antisymmetric differential operator 0 1 (2.II.6b) 1 0, 9X 2.Ill Remarks on the Validity of the Modulational Equations The previous section provides a quite heuristic argument for a Hamiltonian form (.2.II.5,b) of the N-phase modulational equations. We start with action-angle variables the exact N-phase solution (as guaranteed by complete integrability) the (2N) modulational equations for derive immediately from the action-angle representation. It is this structure in the modulational theory that we focus on in this paper; we have not discussed the validity nor accuracy of this modulational system in approximating the exact N-phase solutions. In fact, the validity of the averaging theory for nonlinear dispersive waves is unknown at present, although the very recent work of Lax and Levermore [27] takes the first crucial step by justifying averaged equations for KdV waves. However, for linear dispersive waves, the theory is well understood; the validity of the slowly modulating wave train ansatz is established through stationary phase approximations of Fourier integral representations. Moreover, certain striking features of the nonlinear modulational theory are already evident in the linear theory (for example, the slow space, time dependence of the action variables , which are constant for both the exact periodic Hamiltonian systems with fixed period and the full whole-line Hamiltonian systems). Thus, the linear theory is quite instructive 96 and motivates the nonlinear structure; we now illustrate some of the features apparent in the modulational equations C2.II.5) with the linearized sine-Gordon, or Klein-Gordon (KG), equation; U^^ - U +U = 0, -oo 2ir In the class of periodic potentials with fixed spatial period L = ^ the Klein-Gordon equation is a Hamiltonian system, represented in "x space" by r (° l' n (2.III.2a) l-x oj ,u -u/ 6n XX fL —(n1,tt2 +. u 2 +, u 2,.)dx, , (2.III.2b) %.-k 2 x -L or in "Fourier space" by action-angle coordinates {6 ,J } , n n n=—" 0 l' n ye; U) n (2.III.3a) -1 0/ n loi IL=(oJ + 2 y a)J ,0)^ = 1 + n^K^, k=^. (2.III.3b) Loo nn'n L n=l The inverse map is provided by u(x,t). f ; - r i In terms of the action-angle variables. u(x,t) =J~ ^ sin(t + 0Q(O)) 00 I J + J nr-^ 4sin(a3 t + (|) (0) ) cos (nicx + cj) (0),) , , 12ta n ^n, even ^n.odd n=l \| n where 2 1.2 2 03 = 1+n K , n J = [In I +0) U 3 , n 2(0 ' n' n ' n' n e (t) = 0) t + e (0) , n n n ' and e^(0), n - 0,±1 ,±2,... are uniqualy dataminad by Initial conditions. In the class of potentials on the whole-line which vanish rapidly at Ixl = " , the Klein-Gordon equation is again a Hamiltonian system. represented in "x space" by u 0 11 / (2.III.4a) init H = I |cn^ + u^^ + u^) dx , (2.III.4b) ' •-00 98 and in "Fourier space" by t " \ 6H 0(k)^ 0 1 /S0 (2.III.5a) U(k)i Oi • (SH, ^ V ^SJ / H(k) = I u)(k)J(k)dk . (2.III.5b) 'o Thus, (6(k), J(k)) are action-angle variables for the whole-line KG Hamiltonian system. The map from "Fourier space" to "x-space" is given by u(x) = u(k)e^^dx, n(x) = —^ n(k)e~^'^ dx. /ITT The full wave u(x,t) on -» < x < <» then admits the action-angle Fourier representation: u(x,t) = -p [ sin(kx + a)t +0^(-k)) dk /iTlAr ^J ,o (2,III.6a) _1 sin(kx-ait+e^(k)) dk , A" where u^Ck) = 1 + k^ (2.III.6b) J(k) = [|n(k)l2 + a)2(k)lu(k)|2], (2.III.6c) 0(k) = a)(k)t + 0 (k) , (2.III,6d) o and 0Q(k) are fixed by initial conditions. 99 Consider this Fourier integral representation C2.III.6) of the full wave u(x,t). For t » 0 , a stationary phase approximation of this integral representation shows the first term (involving kx + ut) 1 1 decays like — for x « 0 and like — for x >> 0 , while the second term (involving kx - ut) decays like for x << 0 and like — for x » 0. That is, for large time t » 0 , the /F "left-running" wave (kx + tot) dominates for large negative x and the "right-running" wave (kx-uit) dominates for large positive x. We now explicitly display this behavior for one case. For t >> 0 and x » 0 , the full Klein-Gordon wave u(x,t) is dominated by the right-running wave (with v = — = constant > 0): 1 rr f" /y/, X (k) -it(w-kv) + complex conjugate. ie (k) s If we denote S(k) = w-kv, A(k) H e ° ^ tJOO ' dominant behavior is near the "critical wave number" k which is cr given by S ' Since v > 0, there is one stationary point for this right-running wave (and none for the left-running wave), 1^ 17 V cr /l-?V^ /t^TX^ 100 We note S^''(k " > 0, and from the (l+k2 cr well-known stationary phase formula for a single stationary point k^ I (with s"(k^) > 0), / ; -ltS(k^)-iTr/4 -itS(k) ^ A(k)e 4s^ (k ) we find that for t » 0, x » 0, the full wave u(x,t) behaves like i[0„(k,J -3Tr/A] -itS(k _) o cr J(k„) cr u(x,t)A> e , (2.III.7) '.(k^pts-(k^p 0) (k^^) x (We note that for t >> 0 and 1x1 » 0, 3? « 1 t(l-v2)^2 and «1; k(x,t) changes slowly relative to the t(l-v^) A size of k, and therefore k(x,t) (and thereby ai,J) is a slowly varying function of x and t.) Thus, formula (2.III.7) and the analogous result for x « 0 show that for large time t >> 0 and large |x|, the full wave u(x,t) appears locally as a periodic solution of the Klein-Gordon 2ir equation, with x-period ; however, the wave number k , cr 101 frequency tD(k^^), and effective action Jgff - ^g/; s are cr slowly varying functions of x and t. Furthermore, when restricted to slowly varying periodic waves of the form /j ff(X) -V^^OOT ^ ' «=S=c,I.€t) the full Hamiltonian system reduces to the Hamiltonian system K = 9, 6H = 6J effj (2.III.8) is 6K I with reduced Hamiltonian ^ f" (Refer to Section 2.II and Section 2.VII for single-phase sine-Gordon modulational theory.) We emphasize 1) solutions of the reduced system (2.III.8) agree completely with the leading terms in the stationary phase approximation; 2) the action variable decays algebraically in time. In this linear case, the action variables are constant in time for both the full wave and a wave with fixed period; nevertheless, when the full wave is approximated locally by a slowly modulating traveling wave, the slow change in the spatial period induces on algebraic decay in the effective action. It is these features of 102 the linear theory which appear in the nonlinear theory, but the rigorous verification based on inverse scattering theory (to replace linear Fourier theory) has not been carried out yet. In the rest of this paper,, we illustrate this Hamiltonian approach to modulational theory with a nontrivial example, the periodic sine-Gordon equation. To do so, we derive the necessary ingredients; 1) following [14] for the KdV equation, we give canonical variables for the periodic sine-Gordon Hamiltonian system; 2) after reducing to N degrees of freedom, we provide action-angle variables for the N-phase sine-Gordon wave Qhowever, we must admit that the cycles which define the actions are only clearly understood for N = 1); 3) for the N = 1 case, we show the modulational equations have the Hamiltonian structure (2.111.5). We then close with some potentially important N=1-phase results. 103 2.IV Canonical Variables for the Periodic Sine-Gordon Equation Consider the sine-Gordon equation in laboratory coordinates under periodic boundary conditions, with period L: u ^ - u + sin u = 0 , - " < X < 00 (2.IV.la) tt XX u(x+L,t) = u(x,t) (mod Zir) (2.IV.lb) u^(x+L,t) = u^(x,t) = n(x,t) (2.IV.1C) In Chapter 1, we have discussed at length the relationship between this system (2.IV.1) and the Takhatajian-Faddeev eigenvalue problem; 0 -1\ 0 1^ e 0 •77-r + T (u +n) . ^ = '5; (2.IV.2a,b) 1 0/ 4 ^ X ^ L\ \1 0/ 16v^ \0 e"^" U, in this paper, we adhere to the notation of Chapter 1 and assume familiarity with the results therein. The ingredients we need in order to construct canonical variables for the periodic sine-Gordon equation are the Floquet discriminant, A(E), and one additional eigenvalue problem associated with (2.IV. 2a), referred to as the "yi-eigenvalue problem" 0 -1 '0 1 e 0n -5— + y (u +11) 1' = ^, (2,IV.3a,b) dx 4 X ' _ - I ~ . 1 0 il 0/ 16v''jr7 \0 e ^ 104 The "p-spectrum" = p contains countably infinite elements , with limit points at E = 0, <*> (in the same way as we also remark that the p-spectrum does not remain invariant under flow. Now we define the canonical variables. Consider the discrim- inant A(E) and the j member of the p-spectrum, p^, as functionals of the potentials (u,n = u^) = U ; we then have Theorem 2.IV.1 Define the functionals 4 -1 P^(U) = cosh . (2.IV.4a) Q^(U) = Pj, j = 0,±1,±2,..., (2.IV.4b) where the p-spectrim is enumerated V = = •• • -i»o,i,}. Then the following Poisson bracket identities are satisfied; (Qj.V = . with the Poisson bracket of two functionals defined by 105 The proof of this theorem proceeds analogously to one for Hill's equation [lA]; we sketch the proof in Appendix D. Consider the map U = (u(x),II(x)) ($,?) as a coordinate transformation. If the map is invertible, then by Theorem 2.IV.1 the transformation is canonical. Arguing by analogy with Hill's equation [14], we assume invertibility; for the reduced finite degree of freedom case, the inverse spectral theory of Section l.IV can be applied to prove invertibility. 106 2.V Reduction to Finite CN) Degrees of Freedom The second topic in this sine-Gordon illustration is the reduction to finite (N) degrees of freedom. (We remark that the solution of periodic sine-Gordon when all (infinite) degrees of freedom are excited is yet to be worked out; the KdV-Hill's equation case was only recently solved [32].) From Section l.IV, we consider potentials (u,n) for which the simple spectrum J] is fixed by = {E^,j =1,...2N} . (2.V.1) With initial data taken from this class of potentials, the sine-Gordon wave is an N-phase wave train , explicitly displayed in Section l.IV by the "y-representation" and the "e-function representation". Several remarks on these representations are in order. First, the N variables yj(x,t) in the y-representation arise as the zeros of a quadratic eigenfunction of (2.IV.2a) which is polynomial in the eigenvalue parameter E; in Section 2.IV, the elements of the "y-spectrum" arise in a completely different way. Nonetheless, they are in fact identical (up to a uniform multiplicative constant); this assertion is proved in Appendix E. Second, we note that the fixed simple spectrum ^ • »^2N^ provides (2N) real constants of the motion for the periodic sine-Gordon flow (recall the symmetry Ej => E? for complex E^, Theorem l.II.l), of which 107 exactly N are independent. This is readily verified from the 6-function representation and the quasi-periodic properties of the 6-function, which imply that u(x,t) will have spatial period L provided (in the notation of Section 2.IV) + j, "j K • lor j=X These N constraints reduce to N independent parameters. It follows that the "N degree of freedom" sine-Gordon Hamiltonian system. u 0 l|/^\/6u (2.V.2a) nit \-i (2.V.2b) *"Ij (S) = F 1- (2.V.2c) I - ^"2.'' * *' 2N ' has N constants of the motion, thus complete integrabillty and action-angle variables. Before giving the action-angle variables, we state an important formula. Lemma 2.V.2 If = {E^ £2^^}, then A^(E) Ji(=-V = ? 1 + (2.V.3) 16AE L/A^(E)-4 /2N+1 \ 108 where L = spatial period of u(x,t), M = "charge of u(x,t)": u(x+L,t) = u(x,t) + 2irM, 2N P = n = 0 , k=l N Ah n , and k=l {Aj^,..., Ajj} consists of the simple zeros of A''(E) that interlace E^, k=l,...,2N, and which are uniquely determined in terms of ^(S) via the algebraic system A^(E) dE = 0, j = 1,...,N. (2.V.4) /A^(E)-4 "j Vcycle" I Comments: (i) The verification of this formula proceeds analogous to the corresponding result for KdV-Hill's equation [33]. We describe the details in Appendix F. (ii) At present, except for the N = 1 case (see Section l.V), the definition of the "j y-cycle", as defined by the y-equations of section l.IV, is intuitive at best (McKean [31] reports similar trouble). We are currently seeking a parameterization of these y-cycles. (iii) Another useful form of Lemma 2.V,2 can be developed from the ingredients in Matveev [30]. Consider N holomorphic differentials, {dU^,v=l,...N}, as defined in Section l.IV, together 109 with Abelian integrals of the second kind, and whose only poles occur at E = 0, " with resides Res E = 0} = (-l)-'/16 , Res » E = "} = 1 . These criteria, together with the normalization conditions ^ dn, =0, k = 1,...,N , uniquely define 2* The representation of Lemma 2.V.2 can be expressed in terms of these quantities; for example, when N = 1 we compute = U) kHi. in + i E - E* E 4 -E L dE ^ dE ' E2 - E^ , E, Eg A'^(E) (2.V.5a,b) < L/A^(E)-4 2lidU ^ E (iv) These formulas were derived for periodic potentials by period L; however, the products in (2.V.3) and differentials in /Q\ (2.V.5) extend to the almost periodic case, with = {E^,... jEgjj}. 110 2.VI Action-Angle Description for N Degrees of Freedom The Hamiltonian system (2.V,2) with fixed simple spectrum = {E^,,..,E2jg} is 2N-dimensional, The canonical variables of Section 2.IV, 4 -IF (Qj.Pj) = (yj, cosh ^—^j) , j=l,...,N, coordinatize the system. However, complete integrability guarantees an action-angle description, which we provide next. Denote by ^ an N-vector of independent elements of and define, as in [14]» the action variables " PjdQj , j = 1,...,N, (2.VI.1) "j y-cycle" 2 _1 d]x ^ ttL J 2 y. » II • til 1 II ^ J y-cycle Writing .-1 ^ A^(u)dy cosh —2*^ E. /A2(y)-4 2 Ill then integrating by parts, yields A^(y.) J.(E) = — Jlnvi. (2.VI.2) J Tf ^ L/A2(y )-4 ^ "j%-cycle" ^ where the boundary terms vanish by the algebraic system (2.V.4) which determines the simple zeros of A''(E). We then use the product formula of Lpnmia 2.V.2 which gives N 3.(t) dm, I IH-Vrt::——I——J—^ du., (2.VI.3) V ^ ' / 2N+1 "i%-cycle" ' ^ I n (E-E, ) k=l with all ingredients defined in the lemma. One uses a generating function S(y,?) (if we denote f ^ -J . J (f) = A F (y ,J)dy., then S(ii,J) = I F(p,E) dli) J J J J J "y .-cycle" to adjoin to each action J., an "angle" 0. E —r— S(y,J). By the J J J canonical nature of the map (^,?) (^,J), the dynamical equations become (2,VI.4a,b) 112 where the sine-Gordon Hamiltonian is independent of the angle variables (see, for example. Appendix D), Note that (2.VI»4a) gives no new information, since the actions Jj depend only on the invariant simple spectriim We emphasize that for our purposes, though, the advantage of the action invariants rests with their definition as a loop integral, which stabilizes them to perturbations (at least for the N = 1 kink train). Formula (2.VI.4b) gives a method to compute the temporal frequencies. To explicitly derive w from (2.VI,4b), we must understand the dependence of on the action variables, with given by (2.V.2b), dx + n^) + (1 - cos u) /qS To express this Hamiltonian in terms of I j one uses asymptotic calculations to establish the "trace fomula" [39,45] where the coefficients C^, C_^ are defined by |A(E)exp[i(Y^ - L]) 'c c -r + — +... , Im(E^) > 0, |E1 -> «> ^ E + C_^E^ + C_2E+. Im(E^) > 0, |e1 0. 113 In Appendix F we give a heuristic motivation of these formulas using the "scattering representation of A(E)", Theorem l.III.l: as yet, we have not extended the derivation from compact support potenti£\ls. The end result of that computation is the formula . .M+N+1 = 1 + i- p'j N . . 2N . + I (8A + -j—) -f I (8E + -|—) . j=l J 8 A. j=l J S'^E. J J Using this expression for the definition (2.VI.3) of the roots = 0, N independent members J of and the chain rule, it should be possible to compute an algebraic system for the frequencies o). However, the computation presently eludes us. In the case of the periodic Toda lattice [14], the procedure is tractable due to a formula which expresses the Hamiltonian as a function of the N simple periodic eigenvalues. Presumably the N periodicity constraints stated in Section 2.V yields the analogous formula. The modulational theory of N-phase sine-Gordon waves then proceeds, according to Section 2.II, directly from these action-angle coordinates. In the next section, we give a concrete illustration of this approach with the N = 1-phase sine-Gordon traveling waves. 114 2.VII Method of Averaging : N = 1 Case In this final section, we consider the single-phase averaging of the sine-Gordon equation, with three purposes in mind: 1) to provide a nontrivial illustration of the Hamiltonian structure in the modulational equations, as prescribed in Section 2.II; 2) to announce a potentially important fact for the exact integration (in the sense of Riemann invariants) of the modulational equations for periodic N-phase solitons; and 3) to indicate possible contact of N-phase sine-Gordon averaging with the complete KdV modulational theory of Chapter 3 and [13]. We consider the explicit case where the local wave appears as a "kink train" with sub-unit phase speed (|^| < 1). This two- parameter sine-Gordon wave has been thoroughly analyzed in Section l.V from both an "effective oscillator" approach (with parameters U = ^, E) and from inverse spectral theory (with parameters J] = {£^^,£2}). Excitations "nearby" this periodic traveling wave act as perturbations on the local wave which slowly modulate its parameters, (E,U) or (£^,£2). The explicit map between these two sets of modulational variables is found in Section l.V: 16/E^E2 - 1 U 16v^^ + 1 (.2. VII.la,b) E 115 Equivalently, we can map from (£^,£2) to "canonical" modulational variables, CJ(E^,E2), <(£^,£2)), where the spatial wave number is given in Section l.V and the action integral J is given in Section 2.VI. For this single-phase kink train, Whitham [41] has derived the modulational equations for CU,E): (2.VII.2a,b) 1 with and /2(E-cos u) du V To deduce a Hamiltonian form of these equations (2.VII.3) we need expressions for the action J and Hamiltonian H^^ of this N = 1 kink train. From Section 2.V, we have 116 J(E3^,E2) ^ «> Anyl— (2.VII.4a) \L/A (U)-4 'ji cycle I j u)] dx. (2.Vil.4b) J — From Section l.V, we note the key fact that the "p cycle" for this kink train is equivalent, as a path of integration, to the canonical "b cycle", and is explicitly parameterized in terms of the N = 1 traveling wave u(x,t) by: (2.VII.5) Using this parameterization of the "p cycle", together with equation (2.V.7) for the action J, we find J(E,U) =-p=W(E) , a-u^ (2.VII.6a) W(E) = ^ /2(E-cos u) du. -17 For Hj^, we simply use the effective oscillator equations of Section l.V to find = 1-E + J = 1 - E + ^ W(E) , (2.VII.6b) /l-U^ U7 Hayes [19] has shovm Whitham's equations (2.VII.2) have a Hamiltonian formulation, which in terms of CK,J) becomes (2.VII.7) thereby providing a concrete illustration of our "canonical Hamiltonian" method of averaging (Section 2.II) with this N = 1 train of sine-Gordon kinks. Whitham also computed the characteristic speeds for this system (2.VII.2), S + which for the kink train are real. The system is therefore hyperbolic, and Whitham gives Riemann invariants Z_^ , which are constant along the characteristic directions S^: We simplify these invariants in the form 118 and after mapping to the spectral variables E^, E2 (with (2.VII.1)), we find r 1 2Z+ e ~ = < ^^^2 (2.VII.8) 1 16E^ Therefore, we arrive at a potentially important Fact: The simple spectrum = {£^,£2} of the Takhatajian-Faddeev eigenvalue problem consists of Riemann Invariants for the single- phase modulational equations of the sine-Gordon equation. We have also verified that the same results apply to the N = 1 case of "plasma radiation"; the main features there are (refer to Section l.V): * (1) ^^>^2 conjugate pairs, E^ = E2 , E^ f E2 ; (2) the "y-cycle", as a path of integration, can be deformed into (2*b + l»a) cycles, as opposed to just one b-cycle; (3) the characteristic speeds are now complex; modulational instability is predicted [41]; (4) nonetheless, the simple spectra E^, E^^ (E^ E^)"k still formally provide Riemann invariants along complex characteristics. /q\ In summary, the N = 1 simple spectrum = {£^,£2} provides Riemann invariants for the modulational equations, both for the stable kink train and unstable plasma radiation cases. Compared with our general N-phase KdV modulational results (Chapter 3 and [13]), where the simple spectrum = {X^,... again consists 119 of Riemann invariants for the N-phase modulational equations, we are led to an obvious conjecture for exact integration Cin the sense of Riemann invariants) of the general N-phase sine-Gordon modulational equations. Of course, in this paper we have not derived the modulational equations for general N-phase sine-Gordon waves; that is presently under investigation. However, along these lines we close with a brief indication from this N = 1 case of an underlying structure which suggests sine-Gordon will extend to the generality of the KdV modulational theory. In particular, the main result of Chapters 3,4 and [13] is the invariant representation of the KdV modulation equations in terms of Riemann surface differentials; various forms of the averaged equations are found by evaluating the differential near <» or the branch points of the Riemann surface, while certain physical parameters (such as wave numbers, frequencies, actions) are defined by loop integrals of these differentials. Here we briefly note a similar representation of the N = 1 kink train action J, J = — ^ dv. (2.VII.4a) ir "y cycle" The key is to view the quantity —^ ^ as a differential L/A^(y)-4 2 on the Riemann surface of R (E) = E(E-E^)(E-E2). We employ the product representation (2.VI.3) (with N = 1, M E 1 = "charge" of the kink train), which explicitly yields 120 J = — A Any dto„ , (2.VII.9) TT Z "y-cycle" <=> "b-cycle" lA^'d where du_ = —is the unique Abelian differential of the —- L/A2-4 f second kind with double poles at E = 0, «>, (with 0)2(E) = daj2) Res oi^\ = -^ , E=0 Res 0)2! = 1 , E=<*> and du)2 is normalized around b-cycles, <1 da)2 = 0 • "b-cycle" Comparison with the invariant representation of Chapters 3, 4 and [13] yields promising similarities. 121 2.VIII Future Considerations The following questions are natural outgrowths of this chpater: (1) Does the Hamiltonian structure \6J X prevail for the general N-phase sine-Gordon modulational equations? (2) Are the (2N) simple spectra £^,...,£2^^ Riemann invariants for these averaged equations? (3) How accurate are these modulational equations? For example, are the N = 2 equations accurate enough to describe breather formation, which is the result of the modulational instability of N = 1 plasma radiation? (4) Is there an invariant representation of the N-phase sine- Gordon modulational equations (as there is for KdV theory. Chapter 3, [13]) in terms of invariantly defined differentials on the 2 2N+1 Riemann surface of R (E) = 11 (E-E, )? k=l ^ An initial step in these investigations, which we have just begun, is to study the N = 2 case in detail. CHAPTER 3 MULTIPHASE AVERAGING AND THE INVERSE SPECTRAL SOLUTION OF THE KORTEWEG-DEVRIES EQUATION 3.I Introduction This chapter provides an historical account of our progress on the modulation theory for exact, N-phase (multiply-periodic) solutions of the Korteweg-deVries (KdV) equation: q - 6qq + q =0. (3.1. ^t ^xxx The intent of this chapter is pedagogical, serving to complement our detailed exposition which has already been submitted for publication; the research paper is attached as Appendix H and should be read as the proper scientific account of our results. The content of this chapter includes first a description of Whitham's results for single-phase KdV modulation theory, which motivated our entire study; then we interpret these results in the language of Inverse Spectral Theory (1ST). From the point of view of 1ST, Whitham's results lead to a natural conjecture for N-phase modulation theory. The remainder of this chapter summarizes the approach and results of our paper (Appendix H). 3.II Whitham's Single-Phase KdV Results Whitham [41 ,42 ,43 ] has developed a general theory for studying the slow variations in periodic traveling-wave solutions 122 of nonlinear partial differential equations (pde). Here we illustrate his "averaging conservation law" scheme with the KdV equation. A slowly modulated single-phase periodic wave train q is a wave which appears locally as an exact, periodic traveling wave solution q - qi(e) = (3.11.1) of the KdV equation - 6qqx + ^xxx ^ - ~ < x < «». (3.11.2) Here q^^CO) is 2Tr-periodic in the "phase" 6, where 3 6 = K - wave number 3^ e = (I) - frequency q^ - mean height of the wave. For fixed values of the parameters (K,a),qj^), the phase 9 is given by 0 = Kx + ojt + 6 , o and q^ is an exact solution of (3.II.2). However, for (3.II.1) to be valid over large scales in x and t, the parameters (K,o),q^) vary slowly and serve to modulate the wave train. (For linear dispersive waves, one can establish the validity of the modulating wave train ansatz via stationary phase approximations of Fourier representations [22 ]; moreover, the very recent work of Lax and Levermore [ 27 ] takes the first step in verifying this ansatz for nonlinear dispersive waves.) To model the slow changes in the 124 parameters (K,a),qj^), Whitham introduces two scales, a fast or "local" scale (x,t) and a slow scale (X,T), and assumes (K,(o,q^) depend upon the slow variables X and T . He then provides a prescription to average out the rapid oscillations in the wave q^ and obtain a first order system of nonlinear p.d.e.'s in X and T for the parameters (k(X,T), a3(X,T), q^(X,T)). We now sketch Whitham's prescription for the single-phase KdV equation as well as his results on integrating the system of modulation equations in the sense of Riemann invariants. Consider a single-phase solution, q^(6), of the KdV equation. It satisfies an ordinary differential equation (ODE) in 6: - 6q^q^ + Uq^ = 0, / (J where ( ) H — ( ), and U denotes the phase velocity, U = . K There are two immediate integrals of this ODE, with each picking up a constant of integration; we find 2 2~ (q()^ = qj - f qj + Bq^ + A; (3.II.3) or changing 6-derivatives to x, ^1 • f ^1 ^'^l (3.II.4) 125 But these equations, (3.11.3,4), are differential equations for the periodic Weierstrass elliptic function,• Thus we easily find an exact, single-phase periodic solution of KdV, depending on these parameters which can be taken as (A,B,U) or (K,a),qj^). Another useful way to illustrate single-phase periodic solutions for KdV is to view equation (3.II.4) as an "effective oscillator" (see Chapter 1): •|(qi)^ + V(qj^) = A, (3.II. 5) 3 TT 2 where the "potential" V(q^) = -q^ ^ ^1 ~ ^ total energy. From the potential energy diagram, Figure 24, we find periodic solutions oscillating between zeros of V(qj^) — A. Figure 24 Potential Energy Diagram 126 We now assume the parameters (A,B,U), which are fixed constants in the exact solution vary on the slow scale ( X, T) and modulate this periodic wave. To describe these modula tions, Whitham averages conservation laws; one needs as many conservation laws as there are prescribed constants in the solution being perturbed. Consider a conservation law in the form where 7^ = density and /^ = flux are functions of ;A,B,U). The first three conservation laws for KdV are: (q), + (-3,2 H- = 0 (^2)^+ (-2,3 - i,/ + . 0 (3.II.6) (i" + k')t- VA =" • Notice all densities and fluxes are polynomial in q and its x,t derivatives, so we only have to average such quantities. For some function of q^ and its derivatives, Q[q^(6;A,B,U)], the average of Q = <•)> is defined by integrating over one cycle of 0 27r (or equivalently, one spatial period L = —), the slow parameters A,B,U are frozen In the averaging. In this manner, the averaged conservation law takes the form (see [13] for details) with We now apply this averaging prescription to the system (3.II.6) and obtain a system of modulation equations for the param eters (A,B,U). We first explicitly compute some averaged quantities to give the flavor. iTT = "i? q^(e;A,B,U)de; * ' Jn changing variables to x, d0 = 0 dx = K dx, and integrating over one spatial period L = —2TT , = j q2^(0;A,B,U)dx. 11 Using the effective oscillator equation, (3.II.5), we multiply and divide by obtain W'V - 2i- ;> :}=== /2[A-V(q^)] This averaging representation extends to almost-periodic functions where L is no longer the period but a moderate length on the x-scale. 128 where the "loop" Integral is over one complete cycle in Figure 24. Also notice that basic elliptic integral W, I /2[A-V(q3^)] (3.II.8) that is. One can compute the averages of all densities and fluxes in (3.II.6) in this manner, and in each case the averaged quantities can be given in terms of the complete elliptic integral W. For example. 2v < dqi •felo /2[A-V(q^)] so that K = . (3.II.9) A Carrying out all these averages, the average of system (3.II.6) reduces to (KWg)^ + (-KUWg + = 0 (tcW^)^ + (-kUW^ + A)jj = 0 (3.II. 10) (k)^ + (-kU)jj = 0. 129 Note that this is a nonlinear system of first-order^p.d.e.'s for the unknowns A,B,U, which appear as coefficients in the elliptic integral W. Writing the system C3,II.10) in vector form, we have + D = 0 . (3.II.11a) X where the (3x3) matrices C,D are given by W W W AA AB AU C = W, W, W, (3.II.lib) BA BB BU W, W, W. UA UB UU -UW -uw^ AA D = -uw. (3.11.11c) -UWba W^-UW33 BU V^UA -™UB -^UU The system (3.II.11) governs the slow modulations in the single-phase solution q^(0). Whitham analyzed this quasilinear system and showed it is hyperbolic, and moreover gave Riemann invariants, Z^, i = 1,2,3, which are constant along the respective character istics. Looking at the coefficient matrices, C and D, one can easily appreciate the nontrivial calculations required to prove these facts; to quote Whitham [41], "if various (nontrival) identities among the second derivatives of W are introduced", the equations can be placed in Riemann invariant form. The Riemann invariants 130 are given in terms of the zeros of the cubic (VCq-|^)-A), and the characteristic speeds, frequency, wave number, etc. are all given in terms of complete elliptic integrals. (In the near-linear limit of small amplitude, 1» two of the zeros of V(qj^) - A coalesce, while two of the characteristic speeds likewise merge and match with the linear group velocity. In this sense, we see that the characteristic speeds are regarded as the nonlinear group velocities [ 4l].) To give a very explicit idea about the calculations needed to find Riemann invariants for (3.II.11), we list the series of steps required in the analysis (none of which is trivial); With -u = /MB , the system (3.II.11) can be written as Cu^ + Du = "O . (3.II.12) t X Step 1; One seeks "generalized"left eigenvectors, i= 1,2,3, which simultaneously diagonalize C and D; that is, satisfies Si.C = a. m. , &.D = g. m,. (3.11.13) 1111 ii Left multiplication of (3.II.12) by yields ^ "i*^=0' = (3.II.14) 1 131 Step 2; Manipulation of (3.II.13) shows the characteristic directions? ^i — , i = l,2,3, are "generalized" eigenvalues, the roots of °i ^i det C - D) = 0, (3.II.15) i whereas the are the corresponding eigenvector solutions of I,-(^C-D) -0. 1 Assuming one can find the roots of (3.II.15), with C and D involving nontrival elliptic integrals, equations (3.II.lib,c), and assiming one can solve for the corresponding eigenvectors then one has achieved the form of (3.II.14), ™ n _io'5 i ds ds at a. 3x ' ^ 1,2,3. X So, thus far one has shown only that the system is hyperbolic, assumxng the — are real and distinct. This much done, one moves a. X to the next Step 3; To prove Riemann invariants exist, and find them, the equations (3.II.14) must be expressed in the form - d~ "i • dJ-'idf - (3.11.16) from which one concludes dT • 1 = 1.2.3. (3.II.17) X 132 However, existence of these invariants is equivalent, from C3.II.16), to the following integrability conditions being satisfied: 9Z ^i- = Pi . i.j =1,2,3, (3.11.18) j where m^j is the entry in m^, is the entry in u, and is an integrating factor. The final step 3, which seeks the integral form (3.II.17) of the differential equation (3.II.14), is known as the Pfaffian problem, a classical problem of differential geometry and differential equations [16]. In summary, Whitham carried out all phases of this program to prove the important Fact; The third-order system of modulation equations for the single- phase periodic solution of KdV is hjrperbolic and has Riemann invariants. 133 3.Ill N-Phase KdV Waves and the 1ST Interpretation of Whitham's Single-Phase Results As we have demonstrated above, the existence of Riemann invariants for this modulated system of three, first-order, nonlinear p.d.e.'s is highly nontrival. One might not normally make a big point of such a fact; when in the rich mathematical context of soliton equations, it is worth attention. Perhaps this "complete integrability" of the first-order modulation equations for KdV is a signal of more extensive structure in soliton theory, but now at the perturbation level. Whitham's results predated the 1ST formalism, which since then has provided exact N-phase KdV solutions, denoted qjj(x,t), that are quasi-periodic in space and are given parametrically in terms of (2N+1) prescribed constants, {q^^, ic^, i = 1 N}; qjj(xjt) = q^j fCQj^ ' where 3 9. = K. the i*"^ wave number, X i i til ^t®i ~ '^i ^ frequency, q^jj E the mean height, and f(9^,... is a special function of the N phases 0^. 134 These exact representations of qjj(x,t) have been developed through the Floquet theory of the Hill operator L [28,33,20,21] ,2 L = 2 + (3.III.1) dx With the N-phase KdV wave qjj(x,t) as potential, the spectrum of L, viewed as an unbounded differential operator on consists of closed intervals (bands) on the real axis which are separated by exactly N gaps in the spectrum (Figure 25). Thus the N-phase KdV solutions, qj^(x,t), are synonomously referred to as N-gap potentials. The (2N+1) endpoints of these bands of spectnim comprise the "simple periodic, antiperiodic spectrum" I^®^[28,33], ' k = 0,l 2N}. These elements of can reside anjrwhere on the real axis; we order them -00 < X. < XT <...< < 00. o J- 2N As qj^(x,t) flows according to the KdV equation, the entire simple periodic, antiperiodic spectrum remains invariant and provides (2N+1) constants of the motion (of which only N are functionally independent). /q\ In terms of ^ , 1ST has provided the following represen tation of the general N-phase KdV wave q^^ [9,11,33]: 135 V Efeie ;•"*< }^o ^3 \ V '^*"^1 » Figure 25 Band Structure of a and "y-cycles", N = 4. 136 N 2N q„(x,t) = A -2 I y.(x,t), ha I \ (3.III.2) ^ j=l ^ k=0 ^ The real functions y^Cxjt) reside in the gaps (see Figure 25), y j(x, t) € [^2j—1' ^2j and satisfy the system _ in 't.x -J - -2i"CA-2 I ) (3.III.3) J 1 with R 2(X) E n (X-X,), (3.III.4) k=0 Actually, the reside on the Riemann surface of R(X), and one must specify on which sheet (and in which gap) each lives. As X increases, a solution y^(x,t) of (3.III.3) travels from X-. T to X„. on one sheet and then returns to X„. , on the other 2j-l 2j 2j-l sheet. We refer to this path as the "j y-cycle" and to represen tation (3.III.2) as the representation of Qjj*" We note here that for the N = 1 case, qj^(x,t) is given by q^(x,t) = (X^ + Xj^ + X2) - 2p^(x,t), (3.HI.5) where vi^(x,t) € [Xj^,X2]. That is, there are three parameters in the representation of given by J = {X^,X^,X2}; 137 these are related in a one-to-one manner to the parameters A,B,U in Whitham's single-phase elliptic function solution. Using known solutions of Hill's equation with an elliptic function potential [44], we carry out the exact maps between the various sets of modulation parameters: {A,B,U}, {qj^,Kj^,cOj^}, We state the results as a Fact; The simple periodic, antiperiodic eigenvalues, of Hill's equation with potential q^(x,t) are Riemann invariants for the hyperbolic system (3.II.11) of modulation equations. Thus, the natural 1ST modulation parameters, the simple periodic, antiperiodic eigenvalues of Hill's equation, yield Riemann invariants for the single-phase wave KdV modulation equations. More over, for the N-phase wave qj^(x,t), the (2N+-1) parameters {q^j, i = 1,...,N} again correspond in a one-to-one manner with the 1ST parameters, = {Xj^, k = 0,1,...,2N}. In this light, the modulation theory of qjj(x,t) will involve a system of (2IW-1) . averaged conservation laws, resulting in a quasilinear system of (2N+1) p.d.e.'s for either set of modulation variables, {q^j, or J, ' Based on our Fact, we have the obvious Conjecture; The simple periodic, antiperiodic spectrvmi = {Xj^, k = 0,1 2N} of Hill's operator with potential qj^(x,t) provides Riemann invariants for the modulation equations of the N-phase KdV solution q^^Cx, t). Although we have been led to this rather tempting conjecture, the method of proof for the N = 1 case, due to Whitham, offers no hint at generalizing to general N phases. In fact, as can be 138 seen from the motivation in Section 3.II, even the N = 2 calculations are quite formidable. There have been some investigations on multiphase averaging of N-phase, quasi-periodic wave trains in references [1,2 ,10], but as we have pointed out in the case of Whitham's work, at that time exact N-phase solutions were not yet worked out. Now we have such formulas, in particular for KdV, Moreover, since 1ST has provided these exact representations for q^^, and since we have conjectured that the natural 1ST parameters play a fundamental role in the integration of the modulation equations, perhaps we can develop a modulational theory for N-phase KdV waves based on the structure of 1ST. This turns out to work, and in the remainder of the chapter we summarize our findings. The details are in Appendix H or reference [13]. 139 3.IV 1ST Approach to Multiphase Averaging of the KdV Equation 3.IV.1 KdV as a Completely Integrable Hamiltonian System and Conservation Laws In this section we sketch our approach and results for N-phase KdV modulation theory; the structure of 1ST is shown to provide the necessary ingredients. First, in order to describe the modulations of the N-phase KdV wave, qjj(x,t), we need (2N+1) conservation laws; an infinite family of conservation laws is provided by the Hamiltonian structure of the KdV equation. In the class of quasi- periodic solutions, the KdV equation is a completely integrable Hamiltonian system, It - 0 If • where the Hamiltonian H is H = H(q) = lim Here the Poisson bracket { , } is {F,G} E lim 140 with the gradients defined by SI = lim 6q)dx. —2L r "-Sqc— L-^ Complete integrability is established through a family {H^,m=0,l,2 } of commuting constants of motion, whose gradients can be defined by the Important formula (Thm. 3.2, p. 224 of reference [33]) SH m+l 1 -5 m = (qD + Dq - "I D"') , m = 0,1,2,..(3.IV.1) 6q 5H c = 1. 6q The first few gradients are 1 j = 0 5H^ = < j = 1 6q q j = 2. The commuting Hamiltonian flows generated by the constants are called the "KdV hierarchy." Note that generates translation, while 20.^ = H is the KdV Hamiltonian. In the class of N-phase potentials qjj(x,t), each gradient Cor. 12.1, p. 257 of [ 33 ], we find the important formula N 2 =0 _2 j 6H. ) /2N , I (f-) . PCS) = / n (1-X,r). (3.IV.2) j=0 ^ P(?) V k=0 ^ 2 For example, comparison of the coefficient of 5 gives the "y-1 representation of equation (3.III.2). Moreover, these gradients provide an infinite family of conservation laws for KdV (see Section III of Appendix H): J, , aH. \ - f 6H. 8H T 2 9H n ls,J + S [«% 2 ^ " »• j = 1,2, (3.IV.3) With this choice of conservation laws, and using the almost- periodic averaging, <7^>T + X = 0' j =1,2,...,2N + 1, (3.IV.5a) 142 where /T 1 rfL 3H <7 > = lim — -r^ dx (3.IV.5b) J L-^ J-L % rL 3H . dH. :Xj> = lim W O.IV.Sc) L-^ 2L -L ^ 3.IV.2 Simplification of the Averaging Integrals; Ergodic Flow on an N-Torus Now, the averaged conservation laws (3.IV.5), together with the (S) "p - 2. representation" of the gradients -r—^ , equation (3.IV.2), % bring us very close to a system of modulation equations based on the 1ST parameters = {X^^, k = 0,l 2N}. Unfortunately though, the averaging integrals involve polynomials in the y^(x,t), and the equations (3.III.3) governing the x-t flow of the offer no help in handling the integrals. However, a second representation of q^^ can be developed by integrating the y equations (3.III.3) with the aid of the theta function (see Section II of Appendix H and reference [13]). In Section II of Appendix H , we then derive a representation q^j(x, t) — q^j(0 (x, t),...,0^(x, t); X ) (3.IV .6) which depends upon N real angles 0^, o 0^(x,t) = K.X + (O.t +0 , •J J 143 and which is Zir-periodic in each angle individually, ...,6 0j + 2ir, 0 .,6^^) = qjj(0^,...,0^^) . We call this representation the representation of These two representations of q^^Cxjt), equations C3.III.2) and (3.IV.6), admit geometric interpretations. Denote the manifold of N-phase waves with fixed simple periodic, antiperiodic spectrum = {Xj^, k = 0,1,...,2N}. This manifold YH. is an N-torus (Thm 5.1, p. 231 of [33 ]). The representation of q^^" coordinatiz'=».s this torus as the product of N actual circles, ^ = (0^,02,... ,0jj) € [0,2Tr) X [0,2"rr)x ... x[0,2Tr) . Each of the KdV hierarchy generates a translation on this torus. Alternatively, the p-variables provide a second parametrization of the torus; y = (U"!^ Ujj) € "1®^ p-cycle" x...x y-cycle" = . Now, we aim to use these geometric notions to simplify the averaging integrals in the modulational equations (3.IV.5). This wave qj^(x,t) may be treated as a point on the N-torus , which moves over this torus as (x,t) vary. We then use ergodicity, assuming the spatial wave numbers incommensurate [4 ], to replace the spatial averages in (3.IV.5) with integrals over the torus as parametrized by the 0-variables; 144 1 rZir f2ir Q[qjj(0;^)]de,...,dej^. (2Tr)^ n J r Since the integrands are more simply expressed as functions of the p-variables, via equation (3,IV.2), we then change variables of integration from the 0-coordinates to the y-coordinates, computing the Jacobian as well. The details are in Section IV of Appendix H. In summary, we finally are able to handle the averaging integrals; in fact, we average the entire infinite family of conservation laws at once and obtain the modulational equations in the form * 'Z'x' Here we use generating functions j = \ ("j") I (p ) •» j=0 ^ J j«0 J with given by equations (3.IV.3); the averaged generating functions are then 3.IV.3 The Role of Abelian Differentials in the Modulational Equations; Summary of Results The next step in our development is to recognize the formal series ^7^ expansions of unique, invariantly 1A5 defined differentials ^2 on the Riemann surface of 2 2N R (y) = n (y-X, ), evaluated in the local parameter 5 near k=0 p = 5 _2 = «*>. (See Appendix H for the definition of ^2 ^^d the connection with <'/'> , <^>-) This is the most crucial observa tion, and leads to the following Theorem which summarizes our results. Theorem. Let n denote the differential defined by- n = (n.) - 12 (n,) . It 2 X Then; (i) Near p = 5 _2 = "» (The entire family of averaged conservation laws appears as coefficients in the expansion of near ) (ii) If n is of the form ^ Cr , m > 4N+2, near y = = », then n E p. (When the first (2N+1) conservation laws are satisfied, so are all the higher conservation laws.) 146 (iii) The modulational equations may be represented in the invariant, compact form = 0. (iv) Near \i - = 0, holomorphic part, where the functions are defined in Section VIII of Appendix H . (The branch point - the member of the /Q\ simple periodic, antiperiodic spectrum T is a Riemann invariant for the modulational equations; that is. n = 0 =>.(|j + |^)X^ = 0 for Jl = 0,1,...,2N, with the characteristic speed given by S„ = -12 (For the N = 1 case, we show exact agreement with Whitham's single-phase Riemann invariant results.) The results of this Theorem therefore generalize known results for the single-phase traveling wave, and moreover, indicate perhaps two general facts. First, it suggests that complete integrability of the Hamiltonian system can induce enough structure into the 147 modulatlonal equations to integrate (in the sense of Ri.emann invariants) their first-order terms. Second, while exact, N-phase KdV solutions have naturally led to function theory on a fixed Riemann surface, the modulations of an N-phase KdV wave lead to the deformation theory of Riemann surfaces. CHAPTER 4 CONSEQUENCES OF THE INVARIANT REPRESENTATION 0=0 4•I Introduction The previous chapter has led to a seemingly general framework for multiphase averaging of completely integrable soliton systems: the invariant representation of the modulational equations -12(^2)2^= 0 in terms of Abelian differentials on the 2 2N Riemann surface of R (X) = 11 (X-X, ). A variety of results have k=0 ^ followed from this representation (see the concluding Theorem of Chapter 3). In this chapter we give more evidence for the general nature of this invariant representation. We show the frequency and wave number vectors, to and K, as well as the action variables Jj of reference [14] are explicitly given in terms of loop integrals of and We also deduce conservation of waves from £2 = 0. Finally, we connect the invariant representation £2=0 with the canonical variables method of N-phase averaging as prescribed in Chapter 2 and reference [15]. 4.II £2 = 0; Frequencies, Wave Numbers, and Conservation of Waves From equations (IL8) of Appendix H, the frequency and wave number vectors of the representation of q^(x,t)" are given by 148 149 (N) TK = -Airi C (4.II.1) TO) = -87ri[A (4.II.2) 2N ^(j) where A = ^ X, • The period matrix T and the vectors k=0 ^ are defined in terms of the holomorphic differentials N ^ - [ I xi-h ^ > i — 1,2,.., 1 -.flij=l "ij " •" RU) normalized by the conditions (the cycles are defined in Appendix H) () ip . = S.. . (4.II.3) J "^3 13 a. 1 ^(j) The vectors are then defined by (^(j))^ = C_; the period matrix T has entries Ty = (4.II.4) We state our first results as Fact 1: <.=-() n., j=l,2,...,N. (4.II.5) J J Fact 2: 0)^ = - 12 (j) n,, j = 1,2,...,N. (4.II.6) 150 The proofs for these Facts are quite easy. We use Riemann bilinear identities to show both sides of equations C4,1I,5), (4,11,6) satisfy the same algebraic systems, (4,11.1), (4.IT.2), respectively; by uniqueness of the solutions (det T 0) of these algebraic systems, the Facts follow. The details are in Appendix G. With these Facts 1,2, we have the Theorem 4.1 (Conservation of Waves) (•^J)^ — ~ 1,2,...,N, The proof of this is immediate; we merely integrate the representation n = ~ 12(£22)jj = 0 around the a^-cycle. Since the a^-cycle can be deformed to not touch the s. n = 0 implies a. J (—(> ~ (~12 '• £2^)^ , 2'X a. a. J J and using Facts 1,2, (.<.)„= (Uj)^, conservation of waves. J 1 J A 4.Ill £2 and the Action Integrals In Chapter 1, we defined a key ingredient of Floquet theory, the Floquet discriminant A(X). For the KdV equation, Floquet theory is applied to the Hill operator, L = - 2 + qjg(x,t), with the dx 151 discriminant A(^) analogously defined [28]. In the class of N-gap potentials, q^^Cx.t), which are periodic in x of period 2L, so that the simple periodic, antiperiodic eigenvalues k=0,1,...,2N+1, satisfy the N commensurability conditions, we have [33] dA "" i jT (x—y/) •gr _ 2 i^i*^^ ^i^ where the X^. denote the roots of = 0 which interlace the 1 dX simple periodic spectrum J : < X2 < ^3 < ^2 X^ <...< X2jg_^ ^2N' The constants X^ are determined uniquely in terms of ^ by the system (see Figure 25 in Chapter 3) N / n (x-x;)dx = 0, j =1 N. /2N~~ ~ Vij-cycle B+l-i Since b.-cycle = 1 u.-cycle (Compare Figures 1,2 of Appendix H), j=l ^ these conditions are equivalent to vanishing b-cycles. Thus, comparison with the definition of the differential £2^ (Section VI of Appendix H) yields the important Fact 3; (X) = —A=== » (4.III.1) 2L n/4^A^(X) where ( / = •|jj-( ). 152 In this class of period 2L potentials, Flaschka and McLaughlin [ 14] have provided action-angle coordinates^^, with the action variables given by J. = - "Ijf <> "II ("2 cosh ^(^•^))dy; •J J y^-cycle a short calculation yields Fact 4: J, = — (4,111.2) 3 T' y^-cycle Thus, the action integrals are also carried by the differential Extending this identification to the class of quasi-periodic qjj(x,t), we can multiply ^ = 0 by —» integrate around the lij-cycle, and find with given by equation (4.III.2), ^ (4.III.3) j T ir ]ij-cycle Now combine (4.III.3) with "conservation of waves", Theorem 4.1, (KJ)^ = j = l,2,...,N. (4.III.4) Together, equations (4.III.3) and (4.III.4) give C2N) modulational equations. We have already seen in Chapter 3 that the N-phase KdV modulation theory consists of (2N+1) averaged equations 12 In [12], the action-angle coordinates are for the "mean-zero" wave, with the constraint H =0. o 153 for the (2N+1) parameters mean-zero constraint, H =0, under which J. is defined, does in fact introduce a o ' 3 constraint on k= 0,1,...,2N, effectively reducing the number of independent parameters to C2N). However, the mean-zero constraint is inconsistent with the developments of Chapter 3 leading to the representation = 0. Nonetheless, the structure of equations (4.III.3), (4.III.4) must be essentially the Hamiltonian form of Chapter 2. (Some problems should arise from the "other" degree of freedom) "j/X C4.III.5) 6H _ (Jj)T - SK. 3 T . So far, we have the "left-half" of this form by the identification of Kj, Jj in terms of cycles of The rest of the identification between cycles of and gradients of the KdV Hamiltonian H with respect to will be given in the next section. 4.IV n = 0 and the Canonical Variables Approach to N-Phase Averaging In this final section we aim to establish yet more evidence for the general role of the Abelian differential ^ and its representation of the N-phase modulational equations, = 0. Motivated by the discussion at the end of Section 4.Ill, we seek 3H identification of -r— , —— in terms of loop integrals of vn,, 0 IC. d«J • Z J J ^2? respectively. The first step in that direction is to show the differential is actually a Hamiltonian series. 154 From reference [33], Corollary 3,1, we have = C2m-1) H 5qjj m-1 Inserting this into , the averaged density generating function, we find <^> = jo (%) ° 1^1 '• Thus the differential expanded near n = 5 -2 = "> is given in terms of the following Hamiltonian series; Fact 5; £2^ i!' . ^ r j=-i 2 near y = ? -2 = » = d?[r^+| H^+l + I H2eV...]. (4.IV.1) Via Fact 5, we now have the means for connecting gradients of the KdV Hamiltonian H e 211^ with gradients of simply differentiate equation (4.IV.1) with respect to K. or J.. We also find the J 2 intriguing fact that the entire KdV hierarchy, H^, m=0,l,..., can be read off as coefficients in the expansion of near <*>; in particular, the KdV Hamiltonian H = 2H2 is found from the 4 coefficient of g . Instead of , j =0,1,...,2N, we now work with ^2 coordinatlzed by the "canonical variables" K^, J ^, j = l,...,N. We hold off on the constraint H = 0 until our calculations are o 9H 3H complete. We now outline our approach for connecting ^3 i 155 with cycles of viJ22» ^2' Hamiltonian series represen tation of £2^, equation (4.1V,1), we get gradients of H = with respect to k , J. as coefficients in the expansion near J 3 » of the differentials , Then we search for the proper j j differentials to compute Riemann bilinear identities and arrive at expressions involving gradients of H and cycles of 11^2> ^2' the details in Appendix G, we discover a(2H„) 9H N+l-J!, Fact 6: — 6 H, =7 " (-12fio) (4.IV.2a) ^ "a P=1 J ^ ^ a P N+l-S, = y 0) (4.IV.2b) P=1 P 9(2H ) 9H r ^ ^ j> y£i2- X. X. ^ p Facts 6,7 give the connections we are after, except for the diagonal sum in (4.IV.2) induced by the relation between b-cycles and p-cycles. To obtain a "canonical form" analogous to (4.III.5), we transform away the diagonal sum by defining new phases, 0^, which are related to the phases 0^ by ^ N+l-S, 0 = y 0 . (4.IV.4a) ^ p=l P 156 Accordingly, we define 'lo^, by N+l+il u C4,IV.4b) u - \ "p- P=1 N+l-A N+l-i, r "ji - <^'x" ° 4 • • (4.IV.4c) p=i ^ p=i i With these variables so defined, we start with the compact form of the modulational equations Integrating around the a^-cycle and summing over p from 1 to N+l-£ gives N+l-i, N+l-Jl I () () -i2n, p=l p=i a X p p From Fact 6 and equation (4.IV.4c) we deduce 3(2H_) 3H Fact 8a: , il = l,...,N. C4.IV.5) (VT=l3jr^- 1 3J a /X To gain the remaining N equations, we start with (n^)^ = (I2n2)x ' -iy multiply by , then integrate around the p.-cycle to obtain TF 157 ?^«1 X U^-cycle y^-cycle By Fact 4. this becomes -12i yn. (4.IV.6) TT y^"cycle , To obtain the right-hand side as a gradient of H = 2H2, we need some transformation properties of y-cycles to b-cycles and ic^ derivatives to derivatives; if we define the (NxN) matrix M by 1 1 .1 1 M = (4.IV.7) 1 1 . 1 o-'b then we have ( ) MrU ( ) (4.IV.8a) y^-cycle ^b^-cycle 3 a = M (4.IV.8b) 3ic £ a That is, M is the Jacobian of the transformation from K to £ and, as we have stated earlier, also maps y-cycles to b-cycles. 158 Picking back up at (4.17.6), —> (J^)T = " ^'^2) ' which by (4.IV.8a) p^->cycle / ^ r-l / -12i = M (j) Tin2 I, f and by Fact 7 b^-cycle ^ _^,3(2H2) 3H M I I ^ , and finally from S/ ^ (4.IV.8b) we have 8(2H,) 3H We state the result as '8(2H_) 3H ^ Fact 8b: (Jj^)T ~3^ - 6H ^ (4.IV.9) We put the two Facts 8a,b together in Theorem 4.2 3(2H2) / 3H \ 1 3J X I ®1 3J^ X 5, = 1,2,...,N. a(2H2) ( 3H \ ' (Jj^)T -6 3K^ |X ^1^ X . These (2N) equations thus are the KdV analog of the canonical variables prescription for N-phase averaging. The hybrid nature of KdV, namely its odd order (2N+1) of modulation parameters, surfaces 159 here by deviating from the exact canonical form prescribed in Chapter 2 for even-order modulational systems. Also, Theorem 4.2 accounts for only (2N) of the full (2N+1) equations. Furthermore, notice that if = 0 were consistent with the constraint H s 0 — o (which is not the case), then Theorem 4.2 would take the beautiful form (with H = 2H2) aH ^"^A^T ~ 1 3J„ 1 X £, 3H Nonetheless, we view Theorem 4.2 as positive evidence for the canonical variables method of multiphase averaging as prescribed in Chapter 2. 4.V Concluding Remarks In this chapter we have shown that for N-phase KdV modula tional theory, the differential is more than a mathematical novelty; it carries important physical information such as wave numbers, frequencies, and action integrals. The invariant represen tation £2=0 of the modulational equations has been shown to imply conservation of waves, and as well, has given strong evidence for the validity of the canonical variables prescription of Chapter 2 for N-phase averaging. Presumably, this invariant representation in terms of Abelian differentials can also be shown to reproduce other approaches to multiphase averaging, notably Whitham's average Lagrangian technique [41] 160 and the method of Ablowitz and Benney [2]. Also, we feel analogous representations exist for other soliton systems, such as periodic sine-Gordon, and should lead to integration, in the sense of Riemann invariants, of the N-phase modulational equations. These are topics for future consideration. APPENDIX A SCATTERING MOTIVATION A.l Infinite-Line Scattering Properties Consider the eigenvalue problem (1.II.2) under vanishing boundary conditions at [x] = ", (l.II.4b,c). The details for this problem may be found in [ 39, 26 ]. The "Jost" solutions are defined in terms of their asjmiptotic behavior at x = + ~, the vanishing boundary conditions on u(x), w(x) reduce (1.II.2) to (with X = >''E) (A.l) A basis of solutions for (A.l) is The Jost solutions g^, i = 1,2, are then defined as solutions of the full problem (1.II.2) satisfying the following boundary conditions at x = +» or - » ; as x ->• + " 1 i ^ / as X ->• + " 161 162 1 e as X •*•-<*> y li 1 e as X -> - » . 1-1 If we now define fundamental matrices for (1.II.2) in terms of these Jost solutions, F(x,X) = G(x,X) = (g]^g2). (A.2) then the scattering matrix, T(X), maps the basis elements g^, g2> which behave like the "free" eigenfunctions at x = -», across the influence of the potentials, into the solutions ?2> which behave like the "free" eigenfunctions at x = + ». That is. F(x,X) = G(x,X) T(X) , (A.3) T(X) = t2i(X) t22('^)j It follows that for X real. T(X) can be expressed in tenns of two complex-valued functions: /a(X) -b*(X)\ T(X) = * , X real. (A.4) \b(X) a (X)/ 163 The realization of a(X), b(X) in terms of reflection and transmission coefficients is illustrated by the following example. Consider an incoming plane wave from x = + ® with unit amplitude, which irf.ll impinge upon the scatterer (the potential), resulting in transmission through to - <» of some parts of the wave while the remainder is reflected back to +<= , From (A.3), we have the following (after dividing by . a(X)); a(X) a(X) Referring to the asymptotic nature of ? , g , we write this i i equation in the asymptotic form: / 1 ^ 1 X.. , , w, 1 l] 16X^^}} b*(X) /ll ^^^^~16X^*^ 11 „ _^le . (A.5) a(X) -i| ® )) a(X) i ® X ? + ® -i(X--j|^)x Since e refers to a left-running plane wave, while i(X-Y|x^ X e is right-running, we interpret (A.5) as the free plane- l] "^^^"16X^^ , wave solution of unit amplitude. I e , launched from X = + » and traveling toward the potential; the term b*(X) i(X--j|j^)x e as X -»- + ® is the reflected part of the wave, a(X) traveling to the right, back to x = + ", while the left-hand side. 164 ' i\ ^ e J is the transmitted part of the wave which a(X) has made it through the potential and is traveling to the left out to X = - ». Due to these interpretations, referred to * as the right transmission coefficient, and = Rjj^(X) is the right reflection coefficient. In the same manner, we can launch the free plane-wave eigenfunction from -» , ^l\ I e 16X^ ^, whxchh- h xs• right-running, and find the left transmission coefficient Tj^(X) = and the left reflection coefficient • (Use g^ + §2 a(^^l*^ following facts then follow about whole-line scattering theory: (i) ~ ~ j is the transmission coefficient, (ii) g^ and thereby a(X) can be analytically continued into the upper-half X-plane, where 1 as X -»• «>, ImX > 0. (iii) a(X) -»• < exp[-|-(u(") - u(-"))] as X -»• 0, ImX ^ 0. (iv) |a(X)|^ + |b(X)|^ = 1, X real, (v) the real X-axis is continuous spectrum and is associated with radiation in the sine-Gordon field. 165 (vi) a(X.) = 0 iff X. is a bound state eigenvalue; J J for Re X. = 0, X. is associated with solitons, J for Re X. =j= 0, (X., - X.) occur in pairs associated J 3 J with breathers. In the general case of vanishing boundary conditions at |x| = (l.II.4b,c), the scattering matrix T(X), as defined by equation (A.3), does not extend to the entire complex X-plane. Rather, T(X) is represented by four complex functions; /a(X) b(X) T(X) = , X € C, (A.6) Vb(X) a(X) where a(X) is analytic in the upper half X-plane, ImX ^ 0, and a(X) is analytic in the lower half X-plane, ImX £ 0. In the overlapping region, Im X = 0, (see A.4) a(X) = a (X), b(X) = -b (X), X real. (A.7) However, for the restricted class of compact support potentials, all of these functions (a,b,a,b) can be analytically continued into the whole X-plane. A.2 Proof of Theorem l.III.l From Theorem l.II.l, A(X) = + Using scattering theory of Section A.l, we have the following "asymptotic" behavior near x^, x^ +L: (with a(X) s X - + , 11 1\ 1 111 (|)_^(x,x^,X)~-2 e 1 lii ® X = X t, „ i/ll -^<''><='-'=0' -i (11 i'OXW (f)_(x,x^,X)'--2 |_^J e ~lil® near X = J 1,1, 1/ 11 ,, 9+(X,X^,X)'^'-2 1 iJ a(X) e "'Zl-il^^® ilM^rTTT 1/ 11 ~ 2 1 i) ® 2 l-i) near x = x +L, o ' -±n\ ia(X)(x-x^) ^ -ia(X)(xH-x^) 'l'_(*»Xo'^^'^"~2lil ® "*"2 l-i) ia(X)(x+x^) ^ ^ -ia(X)(x-x^) + —(ij b(X) e +2 i-i near x = x +L, o from which we find 167 A(X) = (f)^^^(x^+L,x^,X) + (})_ 2(Xq+1',X^,X) = a(X)e"^°'^''^^ (A.8) Now, for X real, equation (A.7) yields a(X) = a (X). Thus we write a(X) = la(X)| a*(X) = |a(X)I and (A.8) becomes A(X) = 2|a(X)| cos [a(X)L - ph a(X)], X real. (A.9) APPENDIX B DERIVATION OF THE DECOMPOSITION FORMULAS In reference [ 44 ]» the 0-function we have defined in the paper is denoted 6^, and there are similarly - defined 0-functions ®1' ®2' ®3' fact, in terms of 6^, the N = 1 sine-Gordon 6-function representation becomes eacx.t) + f;B) u(x,t) = 2i£n ecz(x,t);B) (B.l) 'e^a(x,t);B) = 2i An 02(A(x,t);B) We then show the ratio 6^(il;B)/02(A;B) is given by e^a;B) i 0201+1 + j;B) e^OlB)" e^a+f;B) then we use the infinite-product representation of 3^ in the form 0„(£;B) = (constant) e^'"^ n. (l+e^'^^^"®"^^ E (i+e-27ri(nB-A) z n=i -00 168 169 Combining these facts with formula (l.V.lSb) for i!,(x,t) yields, after some manipulation, _ n 2 . . 1-ie " e -n"l a n=iL a ' 1+ie 1+ie " where a « ic(x-x ) + u)t + 2m7iB . n o Using this infinite-product in the formula (B.l) for u(.x,t) yields equation (1.V.18) in the paper. |_| APPENDIX C PROOF: SPATIAL SYl^METRY <=> SPECTRAL SYMMETRY The following proof is based on a whole-line argtiment by E. Overman, to whom we are grateful. Consider any solution ( (1.VI.4); then define the function ^(x,E) by ip (x,E) = (CI) -i \eT"(%(x,E)' With ( )^= ( ), it easily follows that ^ satisfies the associated eigenvalue problem iu(x) '0 y \ iS . ^ (-^2 + "^nCx) ^2^ (' '('l = 0 16/E (C2) -i 0, . io 1 -^"(x) |u(x) (ijj, + T-n(x) '!',) + ( e - ) '>'2 = ^ ^ 16>^ Looking ahead to the spatial symmetry, we also define $(x,E) = 4>(-x,E) , (C3) 170 171 With this foundation, we now assume, as in Theorem 1,VI.2, sjramietry of the potentials: (u(-x), n(-x)) = (u(x), n(x)) . Simply by plugging into (C2), we find Lp-Trnna CI; The following statements are equivalent. (1°) i|i^(x,E) solves (C2) at E, ij;„(-x,—|-) \ $„(.x, -^) 16 E ^ 16^E (2') solves (C2) at E. (-X, —=-)/ \ (x, ) 16 E ^ 16^E The symmetries in the spectrum are displayed with the Floquet discriminant A(E); recall the "Eigenfunction Representation of A(E)", equation (1.II.12), (with x E 0) o A(E) = <|.^^^(L,E) + (C4) •-)- where (^^(x,E) are the basis for the Takhataj ian-Faddeev eigenvalue problem (1.VI.4) normalized by T+(X=0,E) = (J) , ^_(x=0,E) = (5) . (C5) 172 Thus, (|)^(x,E) are two particular cases of the function (|) above, and following the relations (CI), (03) we define / xo, X •^(x) '}>±^l(x,E) \ ^'+(x,E) (C6) -10 (x) 'f'±,2(*»E) $^(,x,E) = ij)^C-x,E) (C7) We also note the initial conditions, which follow from (C5), 0 \ e ^+(0,E) = , ^ (0,E) = (C8a) V 0 -|8(0) e ' and the boundary condition (L,E) = (C8b) TU(L) We aim to relate A(E) with A(—5—) , where 16 E A(E) = <|)^^^(L,E) + = <('+,i(-L,E) + A(-^) = <{.. , (L, -4") + ,(L, -4-)' 15^E ' 16 E 16'•E 173 The approach is to map from to using (C6), and then connect at E and —^ using Lemma CI. 16 E ,(x,E) From Lemma CI, we know i|;^(x,E) = and '1'+ n(-x, —5") le'^E are solutions of the same equation (C2) ; C-x, —|-) j 16 E / using the initial conditions (C8) at x = 0 , K 2^°' "~2~A (3 16 E ° (° o) 1° l(0, I \ 16 E / I?"" a. we find that the eigenfunctions are proportional: 16 E ±|S(P) = e i|':f(x,E) . (C9) ip ^(-x> —y) / 16 E ' 174 Thus, we compute A(_l ) = 4, (L, —+ <}>_ .(L, -^) 16 E ' 16 E ' 16''E -facL) facD = e^ (L, —^) + e^ ip CL, ^-) 16''E 16^E ^u(L) -|u(0) luCD'^uCO) = e e + ® ® '''+ i(-L,E) i[S(0)-8(L)] i[a(L)-a(0)] From the periodicity of u(x) , u(L) - u(0) = 2ir M, M = "charge of u(,x)". we find A(—5-) = (-1)" A(E), 16 E The argument can be reversed in the following way. Note that Lemma CI holds if and only if (u(-x), II(-x)) = (u(x), n(x)). Equating A(E) and (-1)^A(—implies (SC-L), n(-L)) = (u(L), n(L)) 16 E The result then follows using periodicity of u(x), n(x) . | I APPENDIX D PROOF OF THEOREM 2.IV.1 Our aim is to prove the Poisson bracket relations ^ ^ (D.l) The proof is extremely computational, so we only outline the steps; detailed computations are available upon request. For convenience, we center our coordinate system at = 0. First, we must introduce ingredients not defined in the body. From the "y-eigenvalue problem" [J-i-+A + —H - ^7] ^ 5 , (D.2a) ' dx ^ J J = 0 , (D.2b) define the "yi-adjoint eigenvalue problem" [J ^ + A* + H* - (D.3a) VY 175 176 where the matrices J, A, H are given by Notice that the y-adjoint problem (D,3) is simply the complex conjugate of the y-eigenvalue problem (D.2); it follows that {y, i|;(x,y)} is an eigenvalue-eigenfunction pair for the y-problem iff {y^"'"\ (x,y } is a pair for the y-adjoint problem, with y^^^ = y* , (x,y ) = [i^(x,y)] . We denote the y-adjoint spectrum by . 6?. 6C. We begin the proof by computing , where ^ H S. (D.4) * /- (+) Let y^t y, with eigenfunction i(;(x,y^); then y^ c y , with eigenfunction (x,y*) = [i|^(x,y^)] .* Consider the y-eigenvalue problem (D.2) for the pair {y,i|l^}. Now vary the potential = (u,n) (that is, vary the matrices A and H), while holding the boundary condition fixed. (J-^ + A + I^H - ?.)6^ + (5A + -^ 6H) ^ = 6?.(1 + H)^ . j ^ j ^ j 177 An inner product with (x,y*) = [ij^Cx.Pj)] yields [6A + ^ 6H] i^Cx,y.)) J ^ = 6?.(ii^^"^\x,v*), [1 +-y H]^(x,y )) After defining the normalization constant N by N. = [1 + "I H] i^(x,ii.)) , and writing the inner products explicitly, we obtain 5?. i J i r iu(x). 1, ^ -iu(x), 1, x 6u(x) (D.5a,b) 6?. 611(x) Now, let p., y, 6 y, and denote = 5, = x, J K • J K Then {?,?}= T ^ _ A? is. dx; 6u (Sn 611 6u 178 after insertion of (D,5a,b) and repeated use of two formulas which follow from the y-eigenvalue problem (D,2), 1 -iu " 1 iu ^ A A ® '^2^2 ~ 77 ® loC (D.6a,b) 1 —lu ^ 1 lu A A <•2*1'X + 7^ ® *2*2 - m ° Vi - ^*2*2 - ^*i*v the bracket reduces to ^1^2 ^2^1 ^2^1 ^2^^ {?,?} = dx. '^9 2 dx 4NN (? -? )L Integrating this perfect derivative yields x=L ^1^2 ^2^1 ^2^1 {?,?} = A A r ANN x=0 , which vanishes because and ipj^ satisfy the y-boundary conditions (D.2b). Thus, we have shown = 0 • (D.7) The other two bracket relations involve the "momentum" P., J _1 and thus cosh (—• In terms of Q. = ?. = /pT, we introduce 2 ' O T A(Y ) (D.8a) 179 Since the discriminant A is related to the Floquet multiplier O p by p -Ap+ 1=0 (equation (1.II.9)), it follows that -1 cosh (—2''—) = ilnp(yj). Also, at a -eigenvalue, ^ as defined by the initial condition ^_^(0,vij) > is a Floquet solution with Floquet multiplier pCy^) = ^ „(L,y.), J "^9^ J Let r(x,yj) denote the other Floquet solution at y^, with multiplier „(L,p.)]-1 • Then, P.^ can be rewritten as ^ J J P = -^ £n <1. (L,u.) ; (D.8b) J 3 since we already have , yjp , we must compute 6u^li ^ That computation yields 64>_ oCl-'ViJ 1 ,• •— ,(L,U,) — ' t^ Jo « 1 -.22 + ''2*-2 -,1 ife^" e-i" 1 J J 1 ^dx^ . _-iu. L - (,e r^ With this variation in Jin (|> „(L,vi.), the calculation of ^ 3 A, A the remaining brackets proceeds similar to that for {Q.,Q. }, J except the computations are even more complicated. Generally speaking, one uses "Wronskian identities", which follow from (D.6a,b) and -xu (Vl>x f (V">^1^1 + ^ = ^1^2 (D.6c) . iu / ip J il>j' -IT I — 1" \ 3 ^4 ^ A A = Cj » (D.6d) to reduce the integrands to perfect derivatives. The resulting bracket relations are A A (D.9) = «3k • Now, the canonical coordinates of Theorem 2.IV.1 are related to these by 181 "2 2 P. -1 P. = -5i- = cosh 4 „-i "'"i' 2Qj I =3^ Y ~2 We then compute 3Qj 9Qk 3Q, = 0 ; «rV -1 A A. ^ 3Pj^ ^ aPj^ {Q,,P,} = I L3Q, 3P, 3P, 3$, Ji -' 2Q, •'k Q. Q = ^ I <5.. 6, j £ kil Jk Qv ^ = ^jk ' 3P. 3P, 3P. 3P, J k 1 k {Pj.Pk} I 1 LSQj sr, 9Pj SQj A -P. = I 2 "^jS, kX, a L'2Qj 2Q,, -P, 2 kZ 2Q. m^2Q, cs 00 iH fo •rn t— eg •id AS •n A5 < cy K) < (U < or CM CONNECTION BETWEEN DATE p. VARIABLES AND THE "y SPECTRUM" J We express our thanks to Professor H. Flaschka for his usual helpful suggestions in putting the following work to order. First we derive some background results. Let x(x»x^»E) denote a solution of the Takhatajian-Faddeev eigenvalue problem: 0 -1 0 1 e^" 0 — + — w (E.l) 1 0 dx 4 1 01 16)^ -xu Since equation (E.l) is independent of the initial location x = x^, it follows that T— X (x,x ,E) = X (x,x ,E) 9x o '^x ^ ' o also solves (E.l). Using the basis <^^(x,x^,E) of (E.l), which is defined by the initial conditions at x , o 1^ ^+(x=x^,x^,E) , Xjj (x,x^,E) = a(x^,E) |'_^(x,x^,E) + $(x^,E) |'^(,x,x^,E) , 183 184 Evaluating at x = shows a(x^,E) X (x ,x ,E) = '^x o' o' o 3(x^,E) where a(x ,E) = ,x ,E), g(x ,E) = ,x ,E) O X O O X 0 0 o o Next, suppose xCx,x^,E) has prescribed initial data independent of x^: x(x=x^,x^,E) = with a,b independent of x^. (E.3) Then — x(x »x ,E) = = 0 dx o o (•s + tr) o x=x implies x^^ (x^,x^,E) = - ^(x,x^,E) (E.4) x=x From the relation (E.4), we can use the eigenvalue equation (E.l), evaluated at x = x , to compute 1 -iu(x ) e ,16v^ 1 iu(x^) ° I"r w(x„)X(x .E) - ,16y^ (E.5) 185 In particular, (}i^(x,x^,E) are two solutions of (E.l) with initial data independent of x^. Thus with x replaced by - , respectively, we find - ^^(x,x^,E) = "I w(x^)^^(.x,x ,E) - j—^ e ° -y^)|'_(x,x^,E) b I6y^ (E.6a,b) 3 3x~"^-^*'^o'^^ = (-^ e ° ->^j^^(x,x^,E) -•|w(x^)^_(x,x^,E). o 16}^ Now, we def ine (x^, E), Cx^, E) by $^(x^,E) = ^+CXp+L ,x^,E) , (E.7a) l_(x^,E) = |"_(x^+L,Xp,E) . (E.Ba) We recall from equation (1.11,12) of Theorem 1.II.2 that the Floquet discriminant A(E) is given by A(E) = (j)_|_^^(X^+L,X^,E) + (J)_ 2(XJJ+L,X^,E), SO that A(E) = 4'+^i(Xjj,E) + it follows that ir -at-= "• o ' o From the equations (E.6), we immediately find the following T system of linear equations for the vector function F = (Fj^,F2,F2) , f «-.l. »+.2>'= 186 f = X F, where (E.12a) o T iu(x ,) - -iu(x ) J. e °)Ov 1(E- 1 ) 16»^ 16>^ -iu(x„) X = 2^(v^ -• ) - ^(x^) (E.12b) 16i^ , iu(x ) 2i(vf ^ e ) t \ 16>^ We also note that the components of ? satisfy (F^)^ - F2F2 = - •|-(A^(E)-4), independent of x^, (E.13) where we have evaluated the Wronskian ((|), ,({> o*)" ,) = 1 at x = x, TjX ^ 2 2N Since Date seeks f - gh = n (E-E.), whereas 2 12 here F, - F„F, = - y(A (E) - 4) = C(E,E.) n (E-E,), equation 1 ^ J 4 J K (1.IV.2), we see our solution (Fj^,F2,F2) differs from the Dat4 solution (f,g,h) by the constant term C(E,Ej). Thus, = (f,g,h). C(E,E^) APPENDIX F N DEGREES OF FREEDOM : DERIVATIONS 2 / F.l Infinite Product Representation for A -4, A ; Application to f(E) E , uh'-m-h From Section l.III, we note the behavior in the Floquet discriminant A(E) near the essential singularities at E = 0, " ; (-1)^ 2 cos for E A(E)-' < (F.l) 2 cos (v^ L) for E , where M = "charge" of u(x,t) defined by u(x+L,t) = u(x,t) + 2irM. (We note E = 0 is clearly not a root of A = ± 2.) These relations yield asymptotic distributions near E = 0, " for the zeros of 2 (A -4); that is, the periodic, antiperiodic Floquet spectrum ^ = {Ej^A(Ej) = ± 2} is comprised of two countably infinite mOO I PQ sequences, {E } , {E } with respective limit points at " n=-l " n=+l E = 0, <*> , and the asymptotic locations near the limit points are provided by (F.l): E —, n -> + « (E -^®) ° ' 2 (F-Z) (E~0) . 187 188 2 We note: (1°) the zeros of (A -4) are at most double (a proof is given in Section 8 of [31], where a pathology is noted if the sine-Gordon Hamiltonian gets too large; we assiime, as in [31], that is small enough); (2°) the asymptotic relations (F.l) show that near the accumulation points E = 0, " of ^ , the eigenvalues 2 E become double roots of A -4. With these remarks, we now n 2 derive an infinite-product representation for A -4; the proof is modeled after McKean [31], Section 11, where a rigorous justification of our arguments can be found. Fix the simple periodic, antiperiodic spectrum consistent with the N degree of freedom sine-Gordon potentials (Section l.IV): = {E^, j = l 2N} , (F.3) so that all other ^ are double zeros of (A.2 -4). By remark (2 ), all simple spectra E^^S ^ are isolated, away from 0 and We now place all of ^(including into two disjoint sequences, {E }" , {E , where n - n - n=-l n=-l n^ 2 I L E^/^ C^) as n 116mrI n from which we define vo ^ , (r.W zn—X zn P (E)1 for E's'O ; 00 ' E E - Xl • <^•'^1') P^(E) ^ 1 for E , 189 We order E^, n < 0, so that simple zeros of A^-4 occur first. The noted asymptotic behavior of for n < 0 guarantees convergence of these infinite products in the punctured E-plane, 0 < |E| < <*>. We remark that the eigenvalues E^ € J come in pairs, 2 ^Zn+l ' ^2n' double zeros of (A -4) characterized by = ^2^^' With the simple zeros fixed by (F.3), we place (2k), 0 ^ k ^ N, in 't^ji^n>0 remaining 2(N-k) in , which yields P„(E) = n (1 - n (1 - ^) , (F.5a) n=l ^2n-l ^2n n>k \ -N+k E E E 2 P^(E) = n (1 - (1 - n (1-^), (F.5b) n=-l n<-N+k /q\ With I = {Ej^,. ..,E2j^, E_^,...,E_2(jj_j^j} Then, modulo the A^-4 behavior near E = 0, , the ratio ^ ^ is pole-free in punctured o o E-plane, 0 < |E| < ". We now use the as3miptotic relations (F.l); For E , A -44(cos (>^L) -1) = -4 sin (v^L) Since CO 2 sin(>^L)^ n (1 - as E , n -n 2 and E ~'(^) as n ^ + <», n L _ sin^(i^L) , p ^ C ;r^ for E " . 190 Finally, since P^(E)'^ 1 near £ = <*>, we find A^-4'^C. EP (E) P (E) near E = » , (F.6a) J. o For E 0 , A^-4 -4 sin^ (;— 16)^ Since T T ^ sin( ^ 990) J 16y^ 16 n tt'^E and the zeros E^, n < 0, of P^ behave like E^(lenu) n , it follows (with P (E)1 near E = 0) A^-4'^ C-EP (E) P (E) near E = 0. (F.6b) Z o Combining the asjnnptotic relations (F.6a,b) with the pole-free A^-4 I I A^-4 nature of p p in 0 < |E| < », the function ^ p p— is o CO o pole-free and bounded everywhere in the punctured plane 0 < |E| < <*>. McKean [31] shows such a function must be constant, which yields A^-4 = CEP^(E) P^(E) , (F.7) where P^, P^ are explicitly given by (F,5a,b). 191 Next, we apply the same logic to A'^CE). Note the following facts (most are self-evident from above, refer to [3l])j (1°) (E) has infinitely many zeros, in the same fashion as A^- 4; (2°) for each pair ^ i (simple zeros), there is an intervening simple root, A^, of A''(E) = 0 ; 2 (3°) all double roots of A-4 = 0 are simple roots of t/= 0, so that E^ € (E^, n > k, n < -(N-k)) are simple roots of =0; the asjnnptotic behavior (F.2) therefore applies to the roots of A^ = 0. McKean [31] shows that if Hj^, the sine-Gordon Hamiltonian, is not too large, then (4°) there are no double roots of = 0; (5°) there are no additional roots of A"^ = 0 other than those listed in (2°), (3°). We remark that, in this case, all roots of A''= 0 are determined by : the N constraints C2.V.6) fix / o \ Aj, j = l,...,N in terms of ^ » and the remaining roots of A^ -0 are tied to the double spectrum ' (6") Recall the "spines" in the spectrum a (Section 1,111); the above remarks show that the assumptions (i) exactly (2N) simple spectra and (ii) not too large imply there are no spines of spectrum leaving the real E-axis. It seems an interesting problem to investigate the "other" situation, where is large. for existence of > 0 such that -2 < A(E^) <2, A^(E^) =0 (which yields spines), both from the practical point of view of possible sine-Gordon field configurations and from the function- theoretic point of view. (7°) From (F.l), we note the behavior of near E =0," „ IsinC—\ ./ (-1) !• I 16i€ I i~ 16^ 1— I MarE-O, CF.8) A^^-L near E = «> . We now place all roots of l/ =0 in two sequences, consistent with 2 those defined for A - 4; {e;} E {A^ JL, (F.9a) n=l ^n " V(l^) n ^ ; ^^n^ T " ®-(N-k+l)'---^' (F.9b) n=—1 / / L E = E Irj-T—I as n . n n \16mr/ Then we define I^(E) = n (1 - 1^) = n (1 - n (1 - |-) n=l n n=l n n>k n (F.IO) I (E) = n (1 - ^) = n (1 - ^) n (1- ^) , O n=-l- Cl n=k+l b n<-N^TLT ^ 193 where the products converge by the estimates in (F.9). Also note r 1 for E 0 I '-< (F.ll) oa sxn v^L 1 for E <*> sin(_i I ^o~< CF.12) C, 161^ foj. E ^ 0 . We apply the same arguments as before; first note the ratio J. J. is pole-free for 0 < |E| < ". Then analyze the asymptotic o «> behavior (using the facts above): For E 0 , sin(- (-l)^L \i6y^l 16E I I (F.13a) C3I6E o " • For E ^ "o , sin(>^L) (F.lSb) -•ET • 4 194 The same arguments given for A^-4 yield «2 A^(E) = L(a^ + . (F.14) with IQJIOO explicitly given in (F.IO) and constants. We now apply the results (F.7), (F.14) to verify Lemma 2»V.2 : £(E) = L/A^-4 "2 L(a. + :^) I I ^1 E o " , Lv^ /P P • E o <*• and after concellation of common factors from I , P „ , o,<» 0," 1 k „ N A — (a^+—) n (1 - —) n (1 - vc n=l n n=k+l E 2N E E n (1-I-) n (1 - n=l n n=2k+l N PoR ri=1^ ~===^ . (F.15) ^ ^ ' 2N E n (E - E ) n=l 195 The constants 3^^, determined by the asympotic behavior of f(E) near E = 0, " (see (F.l), (F.8)); As E -> <*> , -L sin(t^L) -i ——^ ' L/A2-4 2iLy^ sinC»^L) while from (F.15) , Pi L»42-4 ^ thus = ~| • sin 16E As E ->• 0 , 32 E^2 L2ii sinj- 116>^, while from (F.15), N fi ^ 1^N n A l/A2-4 E^ '2N n E n n=l .^_l^tH-N+l pJf thus $2 - 22A 2N where P = HE, n=l1 n N A = n A . n=l1 ^ 196 The end result is Lemma 2.V.2 : N f(E) = ? U+-^kr^'^ lA^-4 \ / /2iHi \n=lJ n, (E-En ) 2N N with P - n so, A . n . n-1 n=l F.2 Trace Formula for in Terms of = {E^,.., ,E2jj}; A Scattering Motivation Trace formulas in inverse scattering theory are widespread; here we use the "scattering representation of A(E)",Theorem l.III.l. to motivate a trace formula for in terms of the simple r (S) spectrum i = {E^^,.. • .Egj^}. Refer to Chapter 1 and Takhatajian- Faddeev [39] for the scattering formulas we employ. From [39,45] the periodic sine-Gordon Hamiltonian is given by the trace formula where c^ are defined by the asymptotic behavior of Jin a(E), X = (~"7TT the transmission coefficient, see Chapter 1) 1 a *.A ) Zn a(E) = { (F.17a,b) 00 n T c (»^) = c + c . + c -E+ ... as IEI 0. n=0^ -n 0-1 -2 ' ' 197 In Section l.III we have shown a(E)e as |El ->• <», A(E)«.^ E € [0,") . .M . " 16)^^^ I I ^(-1) a(E)e as |E| 0, so that (F.17) may be phrased in terms of A(E) by 1 c c i;-nA(E)~< E € [0,") -iCy'E - "^o c^)^+... |E| -> 0 Moreover, -±/i L E -)• A(E)'-'< iL (-1)^ E -v 0 implies E -»• <*> (-l)^A E -V 0 ; the end result is ^ - 2 %+ as |E| L '^^/2 E ^ [0,«') (F.18) ,>H-1. ,M c (-l)"'"i , (-1)" , , . 1 (-i + as |E| 0, 32EW V 198 From LPTnma 2.V.2, we have an alternative representation of .y /Q\ 1 in terms of \ : (with T — ) L^2Z4 N -i r " ^ ^ a+ b i£i 1. /2N+1 N (E-EJ^) k=l = tt +|> ^ 1=1 ^ _/ 2N 2N+l-( I E.)E^^+...+ PE E' Thus, for E , / N 1 . (1 - ( ^ A )E"^+...) __AL_zi (1+1)2^ Jri 2N , 1 -( I E )E~-^+...) j=l^ N 2N ( I A.) ( I E.) -i=l ^ -1=1 ^ ^ (1 + |) 1 - 1+ 2y^ ^ E J 2E . ! N 1 2N _ / 1 \ -^ + -i ^2 ^ 2 ^ ^ OF ^2/ 2^ 2e'2 \ J ^ j:,! J \E^2/^ For E -> 0, we compute N n (E - A,) a+h^E===- L/A^-4 ^ ^ I2N+1 J,!, 199 N E n (1 - —) CE + r)(-i)^A ^ 2E'"¥ N N ^)E + 0(E2)) = ^ (r + E) ^^ 2E 2N T „ l-Cl ^)E + 0(En j=l j and with F = , we find 16A f -t ^M+l . T >H-1 (-1) 1 ^ K-D^'^h lN]^_ y i(-l) . ^2 (.i,E. l^A.j 32 L/A^-A 32E^^ 2E' + 0(»^) (F.19b) Comparison of (F.18), (F.19a,b) yields N . N -iL (-l)^V 1 + iL 16 16A j=l j=l -J . 2N . N . (-l)^^^^A C_^ = iL + iL 1 y _ y L. 32 2 /, E. /, A. , j=l J j=l 3 'J inserting these into ^ = 2iL tC.i - 16C3_] . f t.ih-N+1 . /N T 2N \ H - 1 + (-3-) Z_. A. HL - ^ + ^ —+ ^ (F,20) \J=1 3 J=1 J/ 2N N where P = II E., A = H A., j=l ^ j=l ^ M = "charge of u(x,t)": u(x+L,t) = u(x,t) + 2irM, and N is the number of phases (degrees of freedom) in u(x,t), APPENDIX G RIEMANN BILINEAR IDENTITIES G.l Proof of Facts 1,2 We prove both Facts at once. Consider the Riemann bilinear identity with ip. and , where k = l,2 is relevant for Fact 1,2, J K respectively. We use Theorem 10-8 of Springer [38]; following his notation we let By Theorem 10-8 we have (G.l) where c^, C2 are defined by the expansion = (c^ + c^ 5 + C2 5^+...)d5. From equations (4.II.3), (4.II.4), we find (G.2) 201 202 Since £2^, ^2 defined to have vanishing b-cycles, = 0, (G.3) To determine c^, we evaluate in the local parameter 5 near ^A = 5,r-2 = "> N g2CN-k)^^ -2 I C, •^3 k=l / 2N 2 . £.-2 /„n (1-X.? ) near X= S =» J Z=o Si this yields c = -2C. , (G.4a) o j,N ' Inserting these fomulas into the bilinear identity (G.l) gives ,(N) T(-J) = -4TRI \ • (G.5a) a. 3 T(-12| ^2) = -8TRI[AC^^^+2C^^~^^]. (G.5b) a. J Comparison of the algebraic systems (G.5a,b) with (4.11.1,2), together with uniqueness (det T 0), establishes Fact 1,2. I I 203 G.2 Proof of Facts 6,7 First, some remarks. By the independence of the coordinates j = l,...,N, we make full use of 3k^ 3J^ 3K^ 3Jj = 0. Moreover, since the a^, cycles can be deformed to not touch the X, 's, and ^ are functions of the X 's only, we k ' j' 3 j=l K are allowed to interchange integrals around cycles on the Riemann surface and derivatives with respect to the coordinates K., J.. J J We prove both Facts simultaneously; the proof requires two bilinear identities. First, consider the bilinear identity with 3J2, , where ^^2 iT With 311 H the boundary of the normal polygon of the Riemann surface of R 2(li) = n (y-X, ), we have k=o k ' f 3J2, P \ N 3(2^ 3^2^ ynaj = I (G.6) 3L„ P=1 3n P P To simplify the right-hand side, we compute 3fi^ I E 0 since has vanishing b-cycles, (G.7a) 9L„ 204 J"'"' " (G.7b) 3L„ P 0 if = J, . The left-hand side is evaluated by the Cauchy residue theorem inside the normal polygon II; 2iTi [sum of the residues inside II] 3n r ji 2N /• y = 27ri Res{., • J m!2 >+ J Ees{l jjj- • J u!l ) (G.8) a •' k=o a •' We compute these residues separately. 8J2l f n _2 Res{", with local parameter V =5 » S/ T 9H „ „ 3HT . , 3H- 3. o , ^ p2 1 . ^ _4 2 , \ -1 1 = Res d? 2 3L„ 4 ^ 3L„ 8 ^ 3L„ * *' Z5—4^ 5 +0(5 ) . 9H„ Ht 3H 12 1 o 8 3L„ 8 3L(, (G.9a) 3f2i rii 2 Res • ^£^2^ » with local parameter , N+2 N+1 . , I E. Xl I D.X^ .1=1 ^ ^ •2 ^"t J=i -Re.[24 +0(5^ + 0(1) ^ >] • [«k^ k 3L. r2N ] n a,-x^) " «r-V r=o r=o rf'k r?4k 205 N+1 . A N+2 , d\ 1vq 1.^=1 = 2 (G,9b) 2N n r-o r^k Using these equations (G.7,8,9) in the bilinear identity (G.6), we have N+1 N+2 9H„ 3H 2N 3X, I Dj\ O . r „ k UKVIJ Fact A: - -pri 2 _ i Ti 2. + 72 8 9^ 8«13L, BL 2N n (X -X,) r=o r k r?^k yS2, 2iri h ' Kh The second bilinear identity is with R and VTT— , 2 3L^ once again with or K^. N . 9n 3n, (y ^2,)= I [<> £22 • y -op () n,]. (G.io) p=l 8L. 311 a b P P For the right-hand side, we find o ^2 = 0 since ^2 vanishing b-cycles, (G.lla) J b 206 N+l-p ^ _ 3 a (> y () yS2. I () yn. 3L, 3=1 ~ HJ-cycle N+l-p = iir "If- I J. j=l ^ f N+l-p '"I,j=l "h'h = < (G.llb) 0 if ^;.=K£ • The left-hand side is again evaluated by the Cauchy residue theorem: py (V I ^2) = 2Tri[Res{", J n2> 3X1 2N 3J2. r VI + I Res{X ,y — .>]• (G.12) k=o " In a similar computation as for Fact A, we use the appropriate expansions in the local parameters near each singularity and find 3J2, p y _c 3H« - 3H Res{»,y 'J ^2^ " Ia 3L^ 8 ^1 ' (G.13a) N+1 N+2 j-1 3S^- 3X I D.XJ' K iiVii = 2 (G,13b) 3L: ^2 > 9L„ 2N n a ) r=o ^ ^ r?ik 207 Putting equations (G.11,12,13) into the bilinear identity (G.IO) yields I •-! N+2 9H„ BH 2N i 2 1 o r 2 k \j^ Fact B: Zl 24 9L„ 8 "l 3L„ , ^ 3L 2N Z k=o II r=o r?«k N+1-5, I ^ !' °2 " '•H ° P=1 i = < Now, to deduce the desired results, simply subtract Fact A from Fact B; the common term involving the three series drops out (quite fortunately) and we find 3(2H„) 3H N+l-£ Fact 6: <)) (-12^2) "a ^ "a p=i =N+l-£ I p=l "p ' 3(2H2) 3H Fact 7: b -12i - 6H 3K„ 1 3K„ IT APPENDIX H MULTIPHASE AVERAGING AND THE INVERSE ' SPECTRAL SOLUTION OF K.dV. 208 209 LA-UR 79-365 TTTLE: MULTZPaASi; AVERAGINS AND TBE ZHVERSE SPECTRAL SOLUTION OF K.(IV. *1. B. riacch)ca, G. Forest, and **2. D. H. Mclaughlin Departsient of Mathematics and Proqrraa in Applied Mathematics, University of Arizona AUTHOR{S): Tucson, Arisona 85721 and **2. T-Division, Los Alamos Scientific Laboratory, University of Califomia, Los Alamos, NH B7545 SUBMITTED TO: To be submitted to Cosn. Pure Appl. Hath. 'Supported in part by H.5.F. under contract No. HPS75-07530 and in part by U. S. Aany under contract No. DAAG29-'7B-G-0059. *1. On sabbatical leave, academic year 1976-79, Math. Dept., Clarkson College, fotsdam, N.V. CQ **2. Supported in part by U. S. Deportment of Energy during academic year 1978-79 as a consultant to T-Division, Los Alamos Scientific Laboratory. . 15 O ^ BvacctpwwefttuitnlcH.thtptmlithirricepnittfTnniM U^. Gevtrrwmm fftiiFti • nofiwciuirat. reytttyfftt tietmi to Dublith or rtpfochjor tt>» oubiithtd form of tftii eontribt»> Tien, or to tllew oihtn to 0o lo. for US, Goverrment pur* 0) Thi LP« Altmo» Scftntiffe fpontopy ftovtni tftti tht eub- > itihtr kMniify thtt irrielt •§ vvofk otrforrmd unetr th« out* piett of Tht Lr£. Otoanrmnt of Entrgy. LOS ALAMOS SCIENTIFIC LABORATORY Post Office Box 1663 Los Alamos. New Mexico 87545 An Atfirmalive AcCon/Equa) Opportunily Bnployer Form No. 836 H2 UMITCD #TAT« SL NO. 3629 DBPAnTMINT OF CNKMOV CONTHACT W9«O»«KNO. >• aSSTK&C? Inverse seectr&l theory is used to prescribe and study equatiens for the slew modulations of N-phase wave trains for the Kort«weg de Vries (K.dV.) equation, to invariant representation of the mpdulational equa tions is deduced. This representation depends upon certain differentials on a Hiemann surface. When eva- liuited near <» on the surface, the invariant representa tion reduces to averaged conservation laws; when evaluated near the branch points, the representation shows that the simple eigenvalues provide Siemann in variants for the Bodulational equations; integrals of the invariant representaticn over certain cycles on the Rieaann surface yield "conservation of waves." Explicit foxoulas for the characteristic Speeds of the oodulational equations are derived. Siese results generalize known results for a single phase traveling wave, and indicate that cosplete integrability can induce enough structure into the modulational equations to dia- gonaldze (in the sense of Riemann invariants) their first- order terms. I. ErrRODCCTIOX ACT SI.'>MnE.V OF THE RESniTS t.A. The General Problec. The mathematical theory {j3, «0 of exact. iBultlply periodic solutions of the Rorteveg de Vries equation (K.dV.) is essentially a chapter in function theory on the hyperelliptic curve 2K - n (v-A..) . 0 Such exact, Eiultiply periodic solutions rarely occur in applications. Rather, one frequently observes vaves vhich nay be described locally (in space x and time t) as aultiply periodic; however, in order for this description to remain accurate over large distances, the parameters must depend slowly on X and t. For such applications the mathematical frameuork changes from function theory on the fixed hyperelliptic curve R(u) to deformation theory of hyper elliptic curves. Applied mathematicians have developed several systematic, although formal, asymptotic prescriptions vhich describe the slou modulations of an approximate multiply periodic vave. In this paper, we use quasi-periodic spectral theory to implement one such prescription for K.dV. Thereby, we de rive a system of partial differential equations vhich govern the deformation of the "local" Kiemann surface K^(u) > iI(w-XJ.). These modulation equations (written explicitly latex in the introduction) admit a very simple, invariant representation which involves certain normalized differentials on the Riemann surface. On the one hand, this representation suggests new mathematical developments; on the other hand, it clarifies aspects in the physical theory of vave train modulations. The matheaatical Interest arises because a physically natural averaging method, applied to K.dV., leads to very special deformations of the Riemann surface Other equations in the "K.dV. hierarchy" must yield deformations of R(u) with a similar structure. One suspects that this family of deformations of the hyperelliptic curve R(v) will eventually be described in purely geometric terms. (Such questions, although they motivate our study, are not addressed in the present paper.) In the physical context of nonlinear waves, our formulas give the first explicit results for the modulation of K-phase, K.dV. vave trains. Our invariant representation also provides Insight into the well-studied case of Dodulatipg traveling waves (N>1). For example, Che equivalence of different versions of the modulation equations (which has thus far been established only through explicit and tedious calculations) is Immediate from the invariant representation. I.B. Background on Modulated Wave Trains. A slowly modulated periodic wave train is a periodic traveling wave q(x,t) • fClar-ut) whose characteristic parameters (amplitude, wavelength, frequency, etc.) vary slowly in x and t. Such wave trains arise is applications in at least two ways. They nay be generated by a periodic signaling device such as a paddle in a wave tank, or they occur naturally as the far field asympcotic state of dispersive wave motion. In either case, the problem of primary physical interest is the des cription of the slow evolution of the physical characteristics. For linear dispersive waves, the situation is well understood. (See, for example. Chapter 11 of reference [li].) First, one seeks a solution in the form of a slowly modulated traveling wave and uses formal asymptotic methods (K.K.B., geometrical optics, multi-scales) to deduce the behavior of such a wave. The most important restilt of this theory is the concept of group velocity. Physical characteristics (such as the energy density) of a nearly monochromatic, linear, dispersive wave propagate at this group velocity. Secondly, one can use stationary phase approximations of Fourier representa tions to establish the validity of the modulating wave train ansatz. For nonlinear dispersive waves, formal asymptotic methods have also been developed to describe slowly modulated wave trains. As yet, the validity of the formal ansatz has not been established although some progress on this "second problem" has been made with inverse scattering representations [«.»']. One general asymptotic method for periodic nonlinear wave trains is due to Hhitham [si]. Be does not discuss the origin of the wave train; rather, he assumes a wave train is present and deduces its modulations. That is, he considers a wave which appears locally as q = q^^ - U + f^(e) vhere Here the wave q^(S) is periodic in 6 of period 2r and is characterized by the physical paraaeters: K - wave number For fixed values of the paraneters (t:,u,U), the phase 6 is given by 6 • »ac + (lit + e , and is an exact periodic solution of the underlying nonlinear wave equation* To model the slow changes in the physical characteristics of the wave q^, one introduces two scales [a fact or "local" scale (x,t) and a slow scale (X,T)] and assumes that the parameters (ic,u,D) depend upon the slow variables X and T. Hhitham provides a prescription to average out the fast oscillations in the wave and to obtain a first order system of nonlinear p.d.e.'s (in X and T) for the parameters u(X,T}, U(X,T)). Ue are interested in the modulational behavior of N-phase, qtiasi- periodic wave trains. Such waves have been investigated previously in ref erences [liltlo]; however, at that time explicit formulas for the exact N-phase nonlinear wave were unknown. For certain special equations, such as K.dV., inverse spectral theory (1ST) now provides exact descriptions of these N-phase, nonlinear wave trains. Vhltham [4l,Si] has applied his general formalism for a single phase (N>1) wave train to K.dV. His modulational equations consist of a rather complicated set of nonlinear hyperbolic equations for (t:(X,T), u(X,T), U(X,T)). In an amazing computation Involving nontrivlal Identities among elliptic in tegrals, Whltham places his equations in Riemann invariant form and gives explicit formulas for their characteristic speeds. (Similar results were found by Miura and Kruskal [34].) The characteristic speeds of these sodulational equations generalize the concept of group velocity to nonlinear vaves. Indeed, is the near linear limit of saall amplitude |q|«l, two of the characteristic speeds coalesce into the linear group velocity. I.e. Modiilated-K-ohase K.dT. Wave Trains. With this background material in mind, we consider a solution of K.dV., • 6qq^ - q^^ , -« < x < • , which appears locally as an N-phase wave train, ^ ~ "13 + f^(6^,•«. ,6^) , «riiere 't®i • "i • Here 1ST provides the exact wave form q^^ in terms of hyperelliptic integrals and theta functions. (See Section II). We assume that the 2IH'l parameters (K^fU^rU) depend upon the slow variables X and T,. and seek modulatlonal equa tions for these parameters. It seems natural to begin by averaging out the rapid oscillations from the K.dV. equation Itself. l«t < •> denote an average over a typical length 2L, 1 • < q> = 2L J ^ q(x,t)dx . In this averaging procedure the slow variable X is frozen, and the length L is assumed to be long on the local x scale, but moderate on the X scale. In our actual computations, we take a limit as L First, replace the K.dV. equation by 0^ < q > + 0^ < -3q + q^ > " 0 This average provides one equation for the 21H-1 quantities Follow ing reference [ft] we conplement this equation vlth the averages of 2K higher conservation laws in order co obtain 21H-1 equacions for Che 2K+1 variables. <7j > + cj. (formilas for Che K.dV. fluxes and denslcies will be given in Section III.) Next, one might use an explicit representation of the local wave q to compute the averages in (X.l) in terms of the parameters Finally, the compllcaced equacions «;hicb resulc muse be brought into manageable form. Although Vhicham accomplished Chls enclre program for Che N>1 case of a single phase wave, Che general V case seems qulce formidable Co us. Isscead, we exploit Che scruccure of 1ST racher Chan use the wave forms it provides. We use an ergodic assumption to replace the spatial averages < •> td.th phase averages. Then ZST is used to compute these phase averages and co place Che modulacional equacions in an iavarlanc form. I.D. Invariant Represencacion of Modulacional Equations. 1ST begins from Bill's operacor L - -D^ + qjj , D • Sjj . vich Che N-phase wave form as Che pocencial. Floquet theory [S3,lt, for Hill's operator defines 2IH-1 simple eigenvalues which carry Che same physical informacion as Che 2IH'l paramecers (See Secclon IX.) Our idea is to transform dependent variables in Che modulacional equacions from (K^,UJ,U) CO Che simple eigenvalues To accomplish chls cransformatlon, we treat the simple eigenvalues as functions of the slow variables X and T. On the Blemann surface of R(v), 2K rv-(u) i n (u-x.) , k-O we define and as the (unique) abelian differentials of the second kind which satisfy the following three criteria: 216 1) r.j has no poles except at u • <°. ii) Sear u •• C•2 • = —|jdc + holooorphic. •» lil) j n • 0 for 1 •• 1,2,...,K. Here b. denotes a canonical set of b-cycles as depleted In Figure 2. Our results are sucunarlzed In this Theorem. Let H denote the differential defined by n = • Then: i) Hear u • C -2 • (Averaged conservation laws appear as coefficients in the expansion of n at ii) If n is of the form n = [c xc + ...]d5 , B > AN+2, near u • 5 -2 • », then n • 0. (Uhen the first 21^1 averaged conservation laws are satisfied, so are all averaged higher conservation laws.) ill) The modulational equations may be represented as n - 0. 217 (iv) Kear - 0, + a boloBorphlc part, where the functions "•f ' are defined in Section Till. The branch point is a Riemann invariant for the modulational equations; that is JJ • 0 - (3^ + - 0 for t • 0,1,...,2N, with the characteristic speed given by fir- v) The wave numbers k and the frequencies u are given by K " ic, " - f f!. Jfi u " ui„ • - 12 / . (See Section XI for precise definitions of K, U, and the cycles a^). The'representation n > 0 implies which is laiown as "conservation of waves." I.E. Discussion of the Theorem. The proof of these results is conceptually straightforward, but the details are a bit involved. The main steps are as follows: (1) Collapse an infinite sequence of conservation laws into e single conservation lav r j r2 J by means of generating functions - I (. 2 ^ Ty ^ I ^ ) Ky (2) Use 1ST to express T, X in terms of and eertnin phases called "the auxiliary spectnsn" (cf. Section IZ, ZZZ). (3) Replace epace averages respect to a certain meamixe J(u)du^f.>duj^, and compute n - 3^^ -12 • 0 Since is iavarlantly defined, T. • 0 contains different representations of the modulatlonal equations. Expanding H • 0 near u « •> provides the modula- tional equations in the form of averaged conservation laws. Alternatively, expanding •• 0 near the siaple eigenvalues places the eqi»tions In Invariant form. Integration of .1 over "a-cycles" yields conservation of waves; In addition. Integration of (uH) over u-cyeles provides the slow modulation of the action variables of [ih], (See Section Till.) This representation should clarify the equivalence of different modulatlonal prescriptions such as the Lagranglan [ll] and Hamiltonian [HI, H] representations. We esiphasize that the simple eigenvalues which are the natural parameters from the viewpoint of 1ST, are also the Klemann Invariants for the modulatlonal equations. Just as the N-phase R.dV. theory has led naturally to function theory on a fixed Rieoann surface, the modulations of an K-phase K.dV. wave lead to the deformation theory of Riemann surfaces. S-1 ease. In order to give the reader a feeling about the explicit nature of our results, we consider a single phase (N'l) wave. Locally, the single phase traveling wave is q • q^^ • Aq + ~ 2 u (x,t) , where < X^ denote real constants and e [>-^,X2] satisfies r 11,0 r~^ "t.x " " [2(^o-^^1+'-2 > J M K-0 • For modulations, the three parameters X^ are assumed to depend upon the slow scales, Xj • Xj(X,T). To obtain equations for the modulations of (XQ,X^,X,), one averages three conservation laws over one spatial period L: 3J < q > + Sjj < -3q^ > - 0 < 4 q* > + 3j. < -2q^ + 2qq^ + 2 " ° 4 qj> + < - f q' ^ 3q'q^- ^ q^ + Sc^®''^ ° " Rather than work vlth the equations la this form, we introduce the differentials !2.f f;,, and n, and work with the invariant repreaentation 1 ^ 2 2 n • 0. The R1 emarm surface of R(tj) > R (u) = I realized as two copies of the R plane, cut as in Figure The a,b, and u cycles are defined in that figure. The differentials are h' {-i'*")R(u)"" with the constants D and £ explicitly given by 2 , .iii L-'R(ii) D - I" 1 . (1.2a) 2 I ' di) "A, R(V) du ^ p-7^W^2>"] R(w) E H f :^-= (1.2b) ['"2 du R(U) The Dodulational equations take the form , 2 n, X,-12 r , X', ,•]. ^ —i—*U -12 (J -7^ + -'u) I "0 . 2 kio L « /J R(u) where ' = and ' = a^.. Sicpanding •?. near the singularity at X. shows that the simple eigenvalue is a Rieaann invariant for the nodulation equations. with the characteristic speed given by - 2D) n>ese results, as well as the conservation of waves and modulation of the action agree canpletsly with the calculations of Hhitham [3,9]. l.G. Validity of the Modulational Equations. In this paper ve focus upon the structure of the modulational equations. We do not discuss their validity. Kevertheless, one naturally wonders about the accuracy with which the averaged theory approximates a solution of K.dV. It must be stated that this accuracy is unknown at present. One can raise many inter-related questions concerning the accuracy of the averaged theory and the type of approximation it is. We close this In troduction with a few remarks about some of these issues. Very formal multi-scale asymptotic techniques have been used to indicate the accuracy of the approximations. (See, for example. Chapters 11.8 and 14.A of HI].) These formal calculations make two points quite clear: 1. More restrictions are needed in order to insure accuracy when N > 1 than when K > 1. (For example, some form of incommensurability among the wave numbers {e^} must be imposed lAen N >1.) 2. The leading order approximation is not an exact solution of K.dV. (To be an exact solution, the parameters must be constant rather than slowly varying functions of x and t.) As described in Section X, 1ST does provide one necessary test on the accuracy of the N > 1 case. This test checks the self-consistency of the reduction process and should be practical to implement. 222 To essabllsh the accuracy of the averaged theory, one would begin by carefully identifying the class of exact solutions which are being approxi- aated. Physically, this class Is selected by the origin of the tsodulated wave- train. (As indicated in I.fi, this origin is not specified in our formal deriva tion of the averaged equations.) Mathematically, the class is selected by initial and boundary conditions. In any case, each class has its own distinct representation of the full K.dV. wave. For example, if the full wave amplitude vanishes as |x| •», the exact solution is represented by the inverse scatter ing transform. If the full wave is quasi-periodic with H phases, the exact solution will be represented by inverse Floquet theory. In either ease, the problem is to examine the local behavior of the full wave and to extract a modulating wave train from the exact representation of the full wave. This line of thought rather quickly leads to an apparent contradic tion. The R.dV. equation is a completely integrable Hamlltonlan system in both classes of fimctlons (the full waves and the exact wave train subsystems). Tet, the modulational equation Si - 0 Implies that the action variables for the reduced subsystem slowly change in space and time. This apparent contradiction may be resolved by realizing that the leading order approximation is not an exact K.dV. wave for the subsystem, even though it Is a good approximation to the R.dV. wave on the full system. To illustrate this assertion, consider the linear K.dV. wave -Scxx • In the class of periodic boundary conditions of fixed period 2L, this Hamlltonlan system may be represented in "x space" as dx , or in "Fourier space" by Here Che cransfomaclos iron "Fourifer space" Co "x space" is given Che fourler series - -J, q(x) • I e ^ ^ + c.c. , K - , J L The map from q(x) (Jj > 6j) is a canonical cransformacion Co accion angle variables for Che linear K.dV. syscea under periodic boundary conditions. In Che flov generated by the periodic Hamllconlan Che acclon variables J are conscanc in cime. On the ocher hand, in Che class of vanishlag boundary condl- cions (q(x) 0 as {x{ " linear K.dV. Is a Hamllconlan syscem «lch "x space representacion" t^ ax 6q ' ® 2 ^ • and "Fourier space representation" S.5(k) - -^1 ii)(k) - k^ ' 3J(k) 3 j(k) - - .0 , H - f k^j(k)dk . ' 36(k) •'O Eere the cransfons from "Fourier" to "x" space is q(s) - —^ f »'kj(k) eiie(W+'«l ait + J..C. . •'o The map from q(x) [J,e}>ls a canonical cransformacion Co accion-angle variables for Che linear K.dV. syscem under vanishing boundary condlclons at Under Che K.dV. dynamics, chese full accions J(k) are also conscanc in cime. Consider che Fourier incegral represencacion of the full wave. q(x,t) - ^ J A5(k) dk + c.c. rcr t » Oi a stationary phase approxination of this integral representation yields iu/4 q(x,t) = vT; J{k ) + c.c. for t»l, x«0, i'6k t cr vith the "critical wave number" given by Ic - + .' cr 3t Thus, for large time t and very negative x, the full wave q(x,t) appears locally as a periodic solution of K.dV. with period Zk/V. except that this 2 cr—^ oeriod 2ir/k . the freauencv u(k } • k . and the effective action J = ' cr wr cr art Jfk^^)/6 depend upon x and t. Fuxthermore, it can be shown that the full Hamiltonian system, when restricted to slowly varying periodic waves of the form reduces to the Hamiltonian system 5E Veff-^xlt • with reduced Hamiltonian H - / dX , L(X) - 2Tr/K(X) He emphasize 1) solutions of this reduced Hamiltonian system agree completely with the leading terms in the stationary phase approximation; 2) the action variable decays algebraically in time. This dependence of the action upon time arises because the Hamiltonian H, which governs the reduced system. is not the R.dV. Haalltoalan in the class of periodic functions with fixed period. (The periods 2i;/<(,%.) in E vary vlth the slov spatial variable Z.) This slou change Is the spatial period induces the algebraic decay in the effective action. In this linear case, the action variables are constant in tiffle for both the full vave and a vave vlth fixed period; nevertheless, men the full wave is approximated locally by a slowly nodulated traveling wave, the effective action for that wave decays in tiae. In this manner, stationary phase approximations of a Fourier Integral representation can be used to establish the accuracy with which the modulation equations approximate the local behavior of a linear wave. For the nonlinear R.dV. equation we expect to replace these Fourier representations with representations from inverse scattering theory. "Trace formulas" re present the nonlinear wave q in terms of a reflection coefficient rOc), q(x) • f ltr(k)f^(x,k)dk + sum over solltons . This representation Is local in x, and one can ask «^ch parts of the re flection coefficient r(k) dominate this representation of q(x) for x near some XQ. Certainly, one can see formal similarities with stationary phase, but a rigorous theory seems far away. 226 II. PSSUKIKARISS OK CUASI-SERIODIC SOLUTIOKS OF K.dV, In the class of quasi-periodic solutions, the K.dV. equation is a eaapletely integrable Bamiltonian system, =-t" ® If' (° = IT )' (II.1) where the Bamiltonian E is E-H(q) . Here the Poissen bracket {,} is -L with the gradients defined by ST = lim ^ fr' f 4^ • 6q) dx . L* 2L •'-LJ \oq -I CcoiQlete integrability is established through a family {H ID ,in>0,l,2,...,} of commuting constants of motion, whose gradients can be recursively defined by the is^rtant formula, (nim. 3.2, p. 224 of mi). £H 1 * V £BUR BH-l iq D + Dq (11.2) 6q J D j 5^ ' » • 0,1,2 % 6q 1 . For convenience, we list the first few gradients: 1, j"0 SK. q j-1 5q I t3q=^-q^] , J-2 The coiaauting Haailtoiuan flows generated by the constants Bj art called the "K.dV. hierarchy". Note that generates translation, while > E is the K.dV. Hamiltonian. Consider a solution of K.dV. which is quasi-periodic in space with exactly degrees of freedom. We denote this N-phase wave by q^(x,t). Sepresentations of have been developed through an intimate relation between the K.dV. equation and the Sehroedinger operator L, L • - . The spectrum of L, viewed as em unbounded differential operator on £ (-cD,a>), consists of closed interveils on the real axis which are sep arated by exactly N gaps in the spectrum (Figure 1). The 2K+1 endpoints of these bands of spectrum are called the "simple spectrum" C33,ll3 - 0,1,2,...,2N} . The points of the simple spectrum can reside anywhere on the real axis; we order them as -•» < < X, <...< X_„ < «>. As q„ flows accord- 0 1 2N N ing to any of the K.dV. hierarchy, the points in the simple spectrum remain invariant and provide (2N+1) constants of motion (of which only N are functionally independent). ' In the rest of this preliminary section we describe two re presentations of the general N-phase wave both of which are vexy useful in our derivation of the reduced, modulational equations. The first representation of q^j is given by [^ ] K 2N q {x,t) " A - 2 I u.(x,t) , I >» . (II " j-1 J k-0 The real functions (x,t) reside in the gaps, Uj(X,t) E ' ^2j' ' ^ , and 8atis£y the system 1,01 n 3 5 - -2i hu-.-2 I u ) ] (11.4) L i-'j ^ J ,'-j -i with 2K tr (X) = n a-K) k«0 Actually, the {v^} reside on the Rieoann surface of R(X), and one must specify on which sheet (as well as in which gap) cach lives (Figure 1). As X increases, a solution of (XX.4) travels from X.. . to X . on one sheet and then returns to X . . on the other sheet. 25* J. Zj He will refer to this path as the "j u-cycle" and to representation (11.3) as the representation of e^.." Bach gradient, 6K. SE. _1 = _i 6c,, 6o ^ q-q„ also has a simple representation!"; indeed, one has (cor. 12.1, o. 257 of OD) " /.2V 6H n d-u.E') Pii) = J f (1-X 5= (11.5) P(5) V k-O " (?or example, compazison of the coefficient of Z 2 gives precisely (II.3). A second representation of can be developed by integrating the u equations (II.4) with the aid of the theta function, •©•(zjT) = I exp{rit2 (S,2) + (Si.Tm)]} , mcZ 229 where z c C' and t denotes a certain HxH siatrix. One fixes a canoni cal set of cuts on the Riemann surface of R(X) (Figure 2), and defines the holemorrshic differentials, &>. / C, . A" R(X) , i - 1,2,...,K , 3-1 nomalised bs the condition •'a.i "^3 ' 'ii • a. Ihenq^ can be represented as llQ QJJ(X,T) - A + r - 2^ log j^©(Z(X,T); T)] , (II.6) where "'a'i '"ii" 'i: i z(x,t) = -2i {c"" [X-XQ] + 2 [Ac'"' + 2c+ d , (11.7) ^ r • r = -2 i j j-1 •'a. ^ and d is a constant whose e^licit fom will not be required. The advantage of this representation (II.6) is that the (x,t) dependence of the K phases z " (s^,...,Z|^) is trivial (linear). n>is explicit linear dependence of z on (x,t) should be contrasted with the implicit dependence of u • (u^r...,ujj) on (x,t) as defined by the u-system (11.4). The penalty for this sijnple (x,t) dependence is that the nap from the phase, vector z to the wave equation (11.6) , is far more complicated than the map frcm u to q^, equation (II.3). A linear transforaation on the phase vector z • (z,,...,z„) X N will place representation (ZZ.6) is a somewhat more use£t:l foim. The transformation is based upon the "periodicity property" of the theta function: i&^z + - exp {-iri[2(m,z) + (m,TO)]} •©•(S,T). Here denotes the column of the r period matrix and the basis vectors for B*", {e'^'}, have conyonents We in troduce a different basis {f^^', j " 1,2,...,N} by N, and e:roand the phase vector z on this basis, [z - Re(d)] " - Re(d^)j e'^' - e^f'B He will refer to the variables {6^}, the e:^ansion coefficients of the phase vector z on the basis, as "angle variables". These angles have scae useful properties. They are real, linear functions of (x,t), e (x,t) « K.x + Ul.t + e , (ZZ.Sa) :) 3 3 v.'ith the wave number and frequency vectors given by K i -4iri T"^ C'"' , (ZZ.8b,c) i -B.i T-^ [A 231 Moreover, if we coordinatize the phase vector z in representation (11.6) by (e^,... {rather than by . ,2^^)), then we arrive at a representation qj.(x,t) « qj. (ej^(x,t) ,ej.(x,t) ; t) (11.6') Which depends upon N real angles and which is 211 periodic in each ancle individu2illy, 5n'®1'"'"'®j-l'®j ®j+l""'®N' " ' the latter following from the periodicity property of the theta function and the enansion of the phase vector z on basis {f'^'}. He will ceiU this representation (Z1.6') the representation of It should be remarked that the angles (6^,...,6^) are natural phases for averaging and that these angles are.not explicitly found in the literature on K.dv. Finally, it is useful to view these two representations more ' r geoaetrically. Denote by M the manifold of K-phase waves with fixed . ' simple spectrum j'®'. Ihis manifold M is an N-torus (Tha. S.l, p. 231 of [»>!)). the representation" coordinatizes this N-torus as the product of N actual circles, 6 • (6,12 ,e_,... ,e„)N e (0,2IT) X [0,2^) x.. .x t0,2ir) • A( . Each of the K.dv. hierarchy generates a translation on this torus. For example, the K.dv. Bamiltonian (H>2H2) generates the t flov of equation (11.8), while the translation Bamiltonian (H^) generates the X flow of (II.8). Alternatively, the u-varlables provide a second parameterization of the torus M: u • (Uj^,...,Vjj) E "1®" w-cycle" x...x"N^ u-cycle" • M. The angle rariablss i yield the sit^lest realization of A( as a torus and the simplest representation of the action of the K.dV. hierarchy as flews on M.If 3he u variables provide the sis^lest ex pressions for the gradients including itself, as functions on *1. Tnis sisnlicity makes the u-paraseterization very useful in our ccoputations. IZI. A GEKERAL PRSSQiZPSZOM OF TB£ HODUIATZOlQa ESUATZOHS FOR K.dV. Consider a solution of K.dV. i^ch appears locally as an K-phase wave whose physical characteristics change slcwly ever large distances in space and time. Zn order to describe this wave, we intro duce two space (and time) scales, a "fast" or local scale x,t and a r(s) "slok*" scale K,S. nie wave itself is described through the "6-], representation"(ZZ.6'), q = qj, \) , (ZlZ.l) where the simple eigenvalues are asstsued to depend upon the slew scales . « X^(X,T) , (ZZZ.2a) \^ile the angles {6^} are predominantly feist variables as is clear from e e, - K. X : D (ZZZ.2b,c} Of course, the wave numbers{k^} and the frequencies {u^} do depend on the slow scales through their dependence upon the simple eigenvalues. We emphasize that the physical characteristics of q^^, such as its frequencies u. and wave numbers k., are fixed by the simple spectrum r(s 1 ^ I (see (Z1.6)-(ZZ.8)). The slow dependence enters our ansatz through j'®'. Our modulational equations will consist of 2N+1 first order p.d.e.'s in (X,T) for the 2N+1 simple eigenvalues. To obtain the aodiilation&l equations, we consider 2N-t-l conservation laws for K.dv. and adapt an early prescription of Vlhitham for a single-phase, periodic wave to this nultiphase, quasi periodic case. Denote the K.dv. conservation laws by * 'x ° 3 • 1,2,...,2N+1 . He evaluate the fluxes and the densities on the N-phase wave (XZX.l), freexe the SICM variables X and T, and average ever the fast space variable x: V "k. L•"I# • ^ (111.3a,b) -k L • Here the length L is assuned large on the x scale, but moderate on the X scale. In the actual ccnputations, we take a limit as L-x°. ?rom here on, all averages < •> will involve this lioit. The modulation&l equations are then prescribed in terns of ' these averaged fluxes and densities: 'V * ^^3^" ° " 1.2,...,2N+1 , (III.4) We use inverse spectral theory as summarised in Section II to (i) obtain a concrete realization of the 2N^1 conservation laws, (ii) eva luate the averaging integrals, and (iii) place the equations in an invuiant form. 234 He begin with a taewn theotein « corollary of which shows that the gradients fH^/oe provide a family of conservation laws. Linearize K.dv. about a solution q to obtiiir. the linear equation Fr. - 6? c+3 £"0, L t X - xjocJ * with foxaal adjoint [®t - ® "J • 0 • tlll.S) Theeren. Tne gradients 5Hj/6q, evaluated on any solution q of iUdV., solve the adjoint equation. [s^-6q5^+5^] ^-0 Vj-1,2,... (III.S') Corollary. The gradients 6H^/5q, evaluated on any solution q of K.dV., provide an infinite family of conservation laws for K.dv., 0 , with densities 6H T (g) = (HI.6a) j °Z and fluxes 6K. 6H. 6H. X, (q) = -2S rr-2- -6 + 6 q ^ . (111.6b) j ^ sex oq oq oq Verifxeation (ef Theorem). Defwe a function g(x,t,T) as solution of the simultaneous equations . oH • 'x 6= .e 61 X oq Sie consistency condition q^. - holds precisely because {B,Z} » 0. Now . 0^ £l , TT "Jt' " ^ TT '®X ^' OC OCT * while °-Tt - ®X < ft so that - il ^ "x 6q 'ft', oq Hie expression in brackets is precisely the adjoint linear operator ®t' * ^xxx' 2 • 11. we have 2^ - - c, and it must be shown that c > 0. But the left side is a certain polynosial in q,q^,..., without constant term. Ihis polynomial vanishes for q • 0, and so vanishes identically. " Verifieagjen (of Corollerv). Consider the fasiily of K.dV. constants {K^}. By the Theoremi their gradients evaltuted on a quasi- periodic solution of K.dV. solve the adjoint equation, 6E. 6K^ 5H • ^ * g - 6 q 3 r-J- " 0 . 3C XXX oe ^ X oq nie recursion £oxiitula (ZZ.2), £E i+1 SH,. X oq 6q ' can be rewritten to show that the last tern in the adjoint equation is a perfect derivative. 5E fiH. ,, 6E. «H. 6 q 3 •' " 3 6 -6q , ^ + 33 . ^ ^ x 6q X oq oq XX oq Thus, the adjoint equation is equivalent to 6E, 6H . 6H^ -2 3 . ^ + 6o , ^ - 6 +1 't[^] ^ XX oq oq 0 , v^ich provides an infinite family of conservation laws and establishes the corollary. • With this choice for the conservation laws, the oodulational equations l»eccme 3Y 6H (X.) = lim ^1 _6-ri!i+6q^,^''i+i In obtaining the average £lwc < , we have used the fact that • 0 , which follows since quasi-periodic functions are bounded. ZV. SZKFI^ZaVTZOK OF TEE AVERASXKS ZRTBGiaLS Using inverse spectral theory, we convert the averaging integrals into a form trtiieh explicitly displays their depen dence upon the sis^le spectrum. Let Q denote either a flux or a density, and consider its average. (Zn this averaging procedure, the slow variables X and T are frozen.) Recall that the wave c[|^(x,t), as a function of the fast variables x and t, is an exact, quasi-periodic solution of K.dV. with N degrees of freedom, nierefore, this wave q^(x,t] may be treated as a point on an N-torus, a point which moves over this K-torus as (x,t) vary. Zn particular,-the variable of integration x in the averages (ZV.l) may be considered as a flew parameter for translation, qj,{XQ,t) qjj (Xjj + X,t) . Since translation is one of the K.dV. hierarchy, this flow generates a translation on the N-torus which is represented by the x flow in (ZZ.Sa) (x,t) « <^x + + e 238 Ihis £la«f will cover the vur£ace of the torus ergodically oroyided the spatial wave ncnbers {le^} are lnc"''™"8urate l4 ]. Oiis avgodi- city permits the replacaaent of integrals over the spatial variable x with integrals over the toms as parameterized by the ? variables: 1 Q> = lijn J dx iin f2-n ... J CiqH«'S»t))de^...dejj (2ir)* J Since the integrands are more sis^ly ea^ressed as functions of V than as funetians of 6, we change variables of integration frcn the 'S^-paranete]Szaticn of the torus to the u-paraneterization: 36 ^e Jacobian 3e/3u is closely related to one coc^uted in (cor. 10.1), p. 248 of [J3]). Its e:5ilicit expression is given in the folloiidng Lemma. The Jacobian 3S/3u is _1 ii. 1 ISirp' 3v " V N (IV.2a,b) n |r(U.)1 i-1 ^ with k-1 V = det du th/ TROT "j -u cycle" 239 With this explicit £OXB of the Jacobian, the averaged equations be COM 8.; + 5j. - 0 , j - 1,2 2N+1 , (IV.3) with (^> -(sS) • where [^2 f 2N (•) K A N 2N-1 n iR(u.)| i-l _ <(•)> = •23 u''-! det r dp "^23-1 R(u) I Although these averages are K-fold integreOs, in the next section it will be shown that, because of the natiue of the integrands, the N-fold integrals reduce to products of single integrals. He close this section with the Proof (of the Iiwinmn). First, we observe f M det J - 2 det - 2" det "j^-u /cycle" ^ b. "2j-l 3 \ for any int«gr«&ds 41^. She first equality is obvious. To verify the last equality, note that f I th / "'i ^ k-j "k -u cycle" ^ " r I „th«, /J '('< S a , kil "k -p cycle" ^ where S denotes a matrix wit^ I's on and below the diagonal, and O's above. So det S • 1, and the last equality follows. We use these relations without further ccBsnent throughout the rest of the paper. d6 To comtjute^ the Jaccbian ou we proceed throu^ the phase variables z: 113p • liaz . ^8v (1V.5) Secall that the angle variables (.6^,...,6^) were defined as the ex pansion coefficients for the phase vector z on the basis ^2ir } , where denotes the column of the T period natrix. Thus, •(k) 1 - s.k * ~- "ke 2ir or equivalently, 2-^B,e«2lIT•to * .to 2, from trtiich we ccopute se. . (1V.6) 241 (H*re> for notational convenience, we have replaced s-Be(d) with s.) To compute the second factor az/du of (ZV.5), we note that Date and Tanaka [^1, in their derivation of representation CZI.6), show that the phase vector z is related to the v~vaxiable by 2 " I I V £-1 where * denotes the vector of normalized holcaorphic differentials of representation (ZZ.6), N Ci - 1 c,^ dv . j-i ^3 3hus, auj. " - R(Uj^) W.7) and the Jacobian (ZV.S) beccBies oy - (Sir)" det IT"^C] det tP] . (IV.e) Since the T-period matrix is given by '«••4 'J -I "i. 4 tfer ®i ""i we find det (T ^C) i fe" 1 • 'b.1 FineJ.ly, we have n (u^-Uj) det (p..)ji - det i'lL).1 —-r I - . (IV.10) n (R(u.)) £•1 242 Placing (ZV.9) and (ZV.IO) into (ZV.e) yields ii. <2?)" l>-i ^ J 3v V N n 1r(U,)| i-l ^ and Mtablisbas the laasoa. o V. SXUPLZFXCATZOH OF SEE ZNTEGIOKDS FTOTii nation of the averaged densities and fluxes frosa equation (1V.4), shows that we aiust average only two types of eaq^essicns, 6H. dH. We will use formal series in order to calculate these averages siBtultaneously for all values of j. First consider the densities fiH ^3 • filN Define T by 243 The representation" of T is given by (IX.S), H 2 a(l-u^5 ) T - (V.3> p(e) Clearly, the sverageddensities can be obtained fren the average of the quantity T. Using (ZV.4c) and writing the multiple integrals as iterated integrals yields • ,2 frSi 1 "" She determinant in the numerator is a polyncnial in (^) of order H> thus, lAere the matrix A( is given by M - - I (V.5a) i 244 and the snatriees h<3) agrae with M except that the column is replaced by the coliam vaetor v, ^i"( Ik'- ^i The steps which reduce (V.4) to {V.4') are a bit tedious; they are described in the appendix. (Equation (V.4') and its proof are very similar to facts in ( l] for periodic q). lbs average fluxes (X. > are acnowhat store conplieatad. r(B) First, we use the "u-], representation** of equation (IZ.3), to write 6H. / K V 6H. In this aiannes, the flux becsnes I r 6H. 6H.1 H SE. \ Next we define - 6( A - 2- ) -2 . / N \ 6H. j-0 "• ( V i-i -^Z To cenpute the average of F, we use (ZZ.5) and (IV.4c) and write ' Since H d_ I dc i-1 £-1 ^ e-0 we have 2 "([i.II-.. II I E-0 (F> - -i2_ PC5) det M Again, the detexninant in the numerator is a polyncoial in (^). Kriting out this polynomial (see the Appendix) yields !523-2 eet I det -12 ( F> J=i 1=1 , (V.7) PCS) det M 246 where the matrices M and are defined in (V.5) and the aatrix agrees with M except that the eolunm is replaced by the column vector v'"*"' , N+l du R(li) (V.B) For convenience, we anomarize. Oie modulaticital equations 3^ with the average fluxes and densities given by the first 2IM-2 terms in the formal aeries H detM!!:^:^ _1 •T,. ! (|if < 1- I det M j-o j-1 P(5) "> /-2.j ,, 1 A N Zn these formulas, the matrix M is given by M - . the matrices agree with M except that the column is replaced by the vaetor V, / H du i RM ' and the matrices agree with M except that the colarsn is •»(+) raplaced hy the vector V , i 1 R(u) ^i 248 VZ. COHKECTZON OF (T> AHD On the Klsnann sozfaee for F.(u) r we define as the unique abellan differentials of the second kind wbidi satisfy the following three criteria: 1) has no poles except at v « 11) Near u » 5 -2• 0. • dE + holenozphie. ill) • 0 for 1 - 1,2,...,H. For j " 1,2 these differentials are of the form N+1 j-1 du 3-1 R(U) ' (VZ.la,b) S+2 . , du I E j-1 3 R(U) Here the constants "^H+l 1/2 S+1 = ^ ' 4 m.c) E^2 = - 1/2 . and the remaining 2N constants D » (D,,.;.,D„)IN and E • (E,IN ,...,E„) are defined by criterion (iii): 249 du 1 / H du RCU) -si" R(U) ' £ N I f „3-l an- E _ i f („N+1. 1 A v") SH- j-1 •i,/ R(u) j 2 J. 2 ' R(P) In the notation of the last section, these linear systeas for D and E beceoe M D - J V , (VZ.2a,b) M S. 1 -A ^ , with solutions "j • £t • 2 S IS ^ i . Thus, the normalizing constants D of differential are related to the expansion coefficients in representation (V.9b) of - - 2 ^ ®N+l-j ^ £!d£ j-0 P(5) 15+1 ,, • , » iZ!ai Oj " - 2 P(5) where C denotes the local eoordlnntev v • € -2 Inserting formulas (VZ.2) yields K 'det I £!S£ (VI.3a,b) ^1- j-1 det M P(5) -. A N A+ J A det r2j "2- 2 2 ^ 2 det M det M r ^ j-i Comparing these formulas for and near y > » vith representations (V.9b,e) of < T> and d£ -2 In addition! » p2 J Siis association of ®t"I - " 3XS " ° • 9)is form of the aodulational aquations will be established in Section VZZ> and its consequences discussed in Section VZZZ. VIZ. A COMPACT FORK OF THE KODaZATZOKAL EQaATZONS • gieorem. On the Siemann surface of RCv), consider an abelian differential 0 whose only singularities occur at X.,/ 0"^*L ''2N* Zf this differential A (i) has at nost double poles at (VIZ.li,ii,iii) (ii) behaves near v • • ® as JJ "• (C„C*'H +...)d5 for M > 4N 2, and (iii) has vanishing B-periods, ^ n • 0 for i • 1,2 H ; then n • 0. Before proving this theorem, we apply it to the averaged equations. Let {X^(X,T), j>0,l,...,2K} denote any solution of the first (2IH-1) averaged conservation laws. Sj ?. = - 123jj£22 . (VII.2) Zbis differential satisfies the criteria of the theorea. Indeed, we cco^te, 2H 0jAj^-12n,X',. N+1 " " ^ Jo " * jJi 'V"')' where • = 3^ and • = 3^ . Equation (VIZ.3] verifies the first criterion because it shows that the only singularities are double poles at (u~^} " ^ vxeept possibly at w • It follows from (VZ.4a,b) that, near U •• C.-2 yj iS. where we have used the facts that • 1 and (X^)• 0. Since the Xj(X,T) satisfy the first (2N-(-l} cooservation laws, we have near -2 n - Z 3j T 1 •'h ® which is the final criterion. Ibus, we have established the Corollary. Zf (3C,T), j » 0,1,... ,211 satisfy the averaged con servation laws h • 0 for i • 1,2 2N+1 , then n i - 123,^02 - 0 • (Vll niis Corollary sunmiarizes two facts: (i) The nodulational equations can be placed in the cce^ct fora (VZX .4)}. (ii) Zf the first (2IM-1) averaged conservation laws are satisfied, then each Btember of the infinite set of averaged conservation laws is satisfied. Since K.dV. does possess an infinite coUeeticn of ceaservation laws, one wonders lAether consistent results will be obtained by averaging any (2IH-1} nesobers of this collection. The Corollary provides strong evidenoe in this direction by stating that the entire collection is consistent with its first (21H-1) members. Zn the remainder of this section, we present a proof of the theorem which follows frca two lennias. T»ittrvi 1. The space of abelian differentials iriiich satisfy the first taro criteria (VZl.li,ii) has ccaplex dimension N. Proof . We use the Riemann-Roch theorem. Define the divisor which has degree d(5) • 0. Following the notation in Reference [38] 254 • we have: L(6~^) = space of aerooozphic functions with at least double zeros at and at nest a 0 2K 4IM-2 fold pole at •. rCS**^) = diaensien of. . X(5) = space of ahelian differentials with at moat double poles at and at leost a (4Ift'2) fold sero at i(5) = dimension of X(6). Then by the Riemann-Iioeh theorem, r(6"^) - d(S) + i{6) - S+1 (Vn.S) Ne show that rCS"^) " 1. Let fc L(£ , and consider g(u). 2N n (p->L) kO .2 Ttae function 9(*) has no finite poles, and near u • C • " behaves as g(5 ) » ^ j «; 4N+2 . If j < 4H42, then by LiouviUe's theorem, g = 0. If j > 4I^t-2, then "1 21J g is constant. 3hus, the space L(5 ) is spanned by jj > and is one diaensional, (r(£ • 1). k«0 From {VZZ.5), we have 1 - 0 + i(6) - N + 1 i(S) • N , which establishes lenna 1. • I fte can vnrite dewn K independent di££exenti&ls Bati«£yiBg the £irst two criteria (VZZ.li,ii): - j~l du J 1 ^ »•, By Iieiwiui 1, a differential !i which satisfies the first two criteria is of the fm N n • I c.Ti^ (711 j-i J 3 for sooe choice of ccnstants {c^}. Lenma 2. If n is a differential which satisfies all three criteria (VZZ.l), then £ n m 0 for i • 1,2,...,H . \ Proof. He use the Siemann bilinear identities [ 16 ]. Define the differentials by Oj • R(u>du , j " 1,2,...,N . TlMn the bilinear identities are I([i;i][V] -[/.;>][/.;])• • where BH denotes the boundary of the normal fozin U [39], and R denotes a differential which satisfies all three criteria. Necessarily, it is of tne foroi (VZX.6). Using this form for it is easy to see that the integrand [A] ° has no pole at ® and at worst double poles, with no residues, at u > for k " 0,1,...,2H. By Cauchy's theorem, the Integral vanishes. thus, we have (Vll I (4, 'j) (/ • ° where we have used the fact that SI has vanishing b cycles, criteria (V.l.iii). Me coB^te that det R(u)dp| ^ 0 The matrix is Invertlble, and formula (VIZ.7) establisbes lenna 2. • Thus, a differential «^ch satisfies all three criteria (VZZ.l) has vanishing a and b cycles. It follows that fl-dF, where F is meroQorphic on the Siemann surface. From the first two criteria, F has at most sisple poles at the and it can be taken to have a sere of order at least 41H-3 at ». Consider the function G > RF. C has no finite poles and a sero of order 4IH-3 - C2N+1) - 2H42 at •>. Hence G s 0, so r = 0, and n E 0. The theorem is established. Vlll. RIEKftKlI HWARIAOTS, CONSERVATIW OF HRVE ACTION, AND ACTION VASIABLES. VTXX.A. RXEHANN ZRVABIANTS At this point, the moduletion&l equations take the coB^aet torm • 0 , (VHI.l) with the differential !2 given by fi -r 3^-12 . Confuting this diffexenti&l explicitly (see VIZ.3) yields 2N C, X^-lZCjAj^ N+1 n-i I 1 k du (U- I (D -12 Jc«0 j-1 R(U) Which shows the singularities of n at u - Expanding n near one of these singularities yields (p-X e o) ' n - i- H ^ (holoBorphic), where N+1 j-1 \- I 2D,-j -t \ = -i-l "1 - 2N n k«0 / 258 N+2 j-1 I 2E X 2K k-O ic*l If the Bodulational •quations 12 • 0 are satisfied, than the singular part of the differential CI near V •• must vanish; that is, Vi " S VJL - 0 for i - 0,1,...»2H (VII1.2) or equivalently, «•= 3-> r • 1 J,'' [t, for Z - 0,1,...,2N , mil.2M where Djj^j " 0 • Equations {3CIZZ.2) place the modulation&l equations in Rieaann invariant form, and show that the ZIM-l points in the single spectrum are Riemann invariants. Qie characteristic speeds are given by ®X • ° (VlII.3a,b) Sj^ = -12 / N+2 j-1 j; N+l 3-1 j-1 ^ ^ In this formula, the constants D - and E • (E^,...E^^) are defined by the Linear systems (VX.2a,b). Solving these systems gives an explicit formula for the characteristic speeds in terms of hyperelliptic integrals: det JUJ du Rtu) S • 6 A-2X^-2 fc, I (VIII.4) det ,j-l du [Ci"'--'" R(U) Clearly these speeds axe real. Qie modulaticoal equations are hyper bolic. Kodulational stability is predicted. Qie significance of form (VIII.3} is that the fizst-order or "streaming" approximation to the full modulational equations for K.dV. is as simple as possible in a fully nonlinear theory. As discussed in the introduction, for the 11>1 case of a single phase periodic traveling wave, Hhitbam placed his modulational equations for K.dV. in Riemann invariant form . Bis derivation uses "various (nontrivzil) identities eunong the second derivatives of VT (p. 569 of Ml]), " ^ iir / t2A-2BiHOn^-2n^]'^ dn. It seems such identities are needed if one were to verify e^licitly that the holcmorphic part of the differential also vanishes. Even for this N^l case we prefer our derivation because the Riemann invariant form of the modulational equations follows from the structure of the differential n. A more explicit calculational approach is suimiiarized in Section IX. VIII B. CONSERVATIOH OF HAVES Next, we descriJse another consequence of the invariant representation Ti 0. This consequence is often called "conservation of waves". Recall the wave nuaber and frequency vectors as de fined in equation (ZZ.8), T K • - 4iri c'"' (VUl.S) T u •-Bid [AC""+2C]. He have the following facts: a) ic • iCj^ - - i b) J - - -12 f {Vlll.6.a,b,c> c) n • 0 "• • 0- ("conservation of waves") To prove these facts, consider the differentials {Sij} and the basis of holoDor^iiie differentials jKl,2,... ,N} as defined above equation (II.6). ^e Riemann bilinear identities [3?] for these differentials are The only singularities of the integrand(in the normal form n occur at V • » «. Expanding near •> yields [/ "i] '"^iiN-l"^ 2 ^ '^is' 3~ *••• ] pj • Using the calculus of residues and the fact that has vanishing b cycles places the Riemann bilinear identities in the form 261 4iri c an I ^ Li i'l • . 3 * 2 ^ "iu' ' 3 • 2 . Ceap«riag the** syctams for If H.) with systaBS (VZZZ.5) tor (ic.u) establlahas facts a and b. Fact c follows by Intagrating & •« 0 over the "a-cyclas". • fWe note facts a and b may be found is [II].) VIII.C. ACTION VRRiaBI£S Consider for a Bonent the special ease where the 2IH1 sismle eigenvalues satisfy the N eccBBensurability conditions iriiieh ia^ly that the wave is periodic in x of period 2L. In this ease, the Floquet diserioinant A(X) [1] satisfies K Si - n (x-x*) 1. dX i-l ^ L 2H n {5i-x.) J k-O where the constants X^ denote zeros of ^ and are given by the system n (X-X!) i-1 ^ dX *0 I 3 * l,2i»»»fN • .th / /2M 3 -U cycle / n (x-x^) V k-O * Since these criteria are equivalent to vanishing b-cycles, we have the identification In thic speciAl case oi period 2L potentials, the action variables in an action angle co-ordinate system are given [1*^] by ^3 J .(fe 3 -u-cycle ^ / A -4 ' thus, we have j -vt-cycle Extending this identification to gaasl-periodic q(*) shows that the action variables J^, which are given in terns of by foznula (VZII.7), satisfy j -u-cycle The equations (VZZZ.6) for "conservation of waves" and (VZZ1.8) for the slow modulations of the action variables, together with an equation for the mean of the wave q^, constitute another equivalent form of the modulational equations. This form of the modula- tional must be essentially the Bomiltonian form of reference CM]; however, a proof requires the identification of cycles of derivatives of the K.dV. Bamiltonian R " ZB^* As yet, we have not found such foanulas. 263 rx. A VERZFZCATZON BV En>LZC3T FOKKDUS In this section we derive the dependence o£ 12 near u ~ Z ' » upon the Rienann invariant fotn of the Bodulational equations. Consider the differential n » 3^(2^ - ** 9iven by equation (VII.3), 2K N+1 . 1 '>,.3-1 5>L I I {D^-12E^) k-0 2 (P-V j-1 R(V) E^^anding near u " " *> and using a geometrical series. i T JiS. .t T- I ^ \ l-\^r 1-0 places (2 in the form n - - I C, £!3£ (IX.l) 3-1 P(C) with the coefficients given by . 2N j-(N+1) JI+2 4.1/ \ 2«Wj-12 * IK 1 < j < M •> •> )c«o =3 ^ (EC.2) 2N j-(H+l) N+2 £-1 / . \ j > N+1 , k-O where " 264 Clearly the nodulatioiud equations (2-0 are satiefied If and only If the coefficients '0. Hotiee that, for j > N^-l, the do not depend upon the quantities (D^-12S^) ; Bather, they depend only upon where is defined by K+2 t-1 " £-1 Note that is the left hand side of the Rieaann invariant equatioDS (VZ1Z.2'); thus, we aee rather e^^licitly that the tail of (2 vanishes pro* vided that the Riemann invariant equations are 'satisfied. To examine the first H coefficients we need a fozBsula for (D.—12£j)< Recall that D. and E. are defined by the noxBializa'> 3 3 J J tion conditions. / , reps, j Jbi b^ Differentiating these conditions yields Titiwar systems for D and E' which can be added to obtiiin a linear system for the cenbination (D-12E*): H(D-12S') - - I" I £. , . . - 6 X/ u ^ Jc-0 JC As already stated, hand side of the Hiemann invariant equations (V2ZI.2'); thus, if these equations are satisfied the integrand on the right hand side of systen (ZX.3] has no pole at V " In any case, we can^write Iben using the fact that is e factor of £|^(u)~fj^(A^) > we place the linear system (IX.3) in the foxa where IH-2 r'» . rf . i;' (•..ij - i2E,xi) Solving for D-12S' yields 2K 1 r du 2(e^-12E')-- I I "S j kmO £•1 ^ •'lbj R(P) Inserting this expression into (IX.2) yields • 2N £ 1 du MC - - I :(V k-O . 266 where C • eoX(Cjj,Cj^^,...,C:^). Z£ the Biamann invariant equationa (VIZ.2') are satisfied, fCX^) " 0; thus, • 0 for i " 1,2,...,N. In this Banner, we use an explicit foxBiula for (2 to verify that if the Riemann invariant equations are satisfied, 12 • 0. X. COKCLUSZON He close this paper with sow ratiarks about the aodula- tional equations and their derivation. First, inverae apeetral theory (ZST) provides a very ayataaatie derivation of the aodulational equations, a derivation lAich begins from the natural starting point of averaged conservation lawa and eenclndes with an invariant repreaen- tation of theae equations, n - - 12 Cj - 0 . (X.l) Siis reduction is aeemnplished through the theory of ZST and its Siemann surface stmeture, rather than through detailed calculations which use the explicit fozD of the H-phase wave. Shis feature of our derivation through ZST is is^rtant because the N-phaae wave form is a very ccoplicated egression iriiich is cumbersoBie to smnipulate. Secondly, we es^hRHiize the precision which ZST introduces into the description of the averaging. Intuitively, it seems clear that one should remove rapid oscillations by averaging over a long length L on the fast scale. The problem is to convert these spatial integrals into a more tractable form. ZST acecsplishes this conver sion by using the ergedic flew on the torus to transfozm the spatial integrals into products of hypcrelliptic integrals, thus ais^lifying the modulational equations. Once these modulational equations have been integrated, pe:Aaps nvBserically, it ahould be possible to devise a practical test of their consistency frcci a necessary condition for their validity: The slowly varying wave numbers {} must be such that the trajectories 6^(x), rather snoethlv cover the torus in a moderate lenerth L. He feel that the invariant repreaentaticra (X.l) of the Bodula- tional equations is both Iseautiful and useful, nte differential C near •J m K provides the eoaservation foxa of ^e equations; the same differen tial near ^ yields their Rieoann invariant term cycles of n and yC] yield "conservation of wave action" and modulations in the action variables. It should be possible to extract from 0 other forms of the modulational equations such as their Lagrangian III] and Bamiltonian [SS,5'f] representations. Future projects for the K.dV. modulatienal equations include (i) a numerical and analytical study of the consequences of their Riemann invariant form, (ii) a nisBerieal study of the accuracy of (X.X), and (iii) the iaproveaent of the modulational equations by the reten tion of higher order derivatives. Ihe numerical studies could begin by generalizing the cooputations of [ A}] from the N > 1 case of traveling waves to the case of wave trains with several phases. She necessary criterion of smooth covering of the torus should also be checked numerically. Our results show that the coo^lete integrability of H.ttV. isposes enough structure on its modulational equations to diagonal ire their first order or "streaming" terms. Often, however, experience indicates that second order derivatives must be retained in the mod ulational equations in order to insure accuracy. Does cocplete inte grability of K.dv. also impose structure on modulational equations whiei) include these higher order corrections? He believe results similar to (X.l) will apply to other equations which are integrable by XSS. For exanple, it would be interesting to consider the sine-Gordon (»qt>ation. For N-1, it is known "kink train" is modulationally stable, while "plasma radiation" is modulationally tmstable. This instability leads to the formation of bound states of two solitons, that is, to "breather" formation. The possible connection of this instability with two phase wave trains seems clear. Mathematically, similar results should be valid when the K.dv. equation is jreplaced by any member of the K.dV. hierarchy. He conjectiae that a wave described by the Bamiltonian system Oj. • {q,H^) will modulate as (see the defiinltios of in Section VI) • constant . Although our derivation is systcaatic, it is also lengthy. For such generalizations, it beccMS far too ceopllcated. (For exiDple, the fluxes for the n.dv. hierarchy beesne unaanagaable.) The sisplicity of the final representation (X.l) indicates that its derivatiOD should be intrinsically connected with the theory of Rleaann surfaces, at least lAen the surface is hyperelliptic. Zn this direc tion, we axe beginning to apply the general Hieaann surface approa^ of ICrichever to modulational problens. In any case, the present paper suggests that nodulational behavior of a ccc^letely iategrable wave equation nay lae as closely related to the defomation theory of Rieoann surfaces as the exact representation of quasi-periodic waves is related to function theory en one fixed Bieoann surface. 269 APPENDZX SVaLlBVTZON OF TWO QBTERUXKaNTS Za this appendix, we evaluate two detexninants tdiieh are used in Section V: r N dett / (l-ay)u^"^-1 du R(v) det M - I det (A.1) j-1 d dat f Cl+cu)a-au)«^-^f^ - lo? o (+) ik.2) de • j"l C-0 nirbughout this appendix, the matrices U, are as defined in Section V. All other notation applies only in the appendix itself. First, consider (A..1). Clearly det det - a BJ •'b with the matrices M and B given in terms of their column vectors by Bt" . . B'*" U 5 , B = . T ' J =.j „3-:1 du R(U) 'i •'b.1. ='"-[5"']. Ifer • i b Exterior product notation [ places the detczninant in the £OZBI det tM-oB] - a A..,A [M""- OB""] ' Consider the coefficient of in this product. It stust contain j-"B-v«ctors" and H-j "M-v»ctors"s bemvar, cooMideratien of the chain of equalities M«). 1'^' m"). |(2) (A.3) „W). 9(H-I) shows that each such term vanishes (because the product contains repeated vectors) except the term AM«' A...AM'«-^' A...A?'«' . Dsing the equalities (A.3) then yields det [/.<-a8] - det M + I A...A A A...A M""Ai'"'] j-1 iK^ finally, moving the vector B j-1 places yields det [M-oB] - det M - A...A A b"" A A...A v^ich estublishes (A.l). Turning to (A.2), we define the matrices CU) B j Cl+eu) (l-aw)u^^ . ^i • ? + e E , «rtiere . r - E- f (l-av)p^ ^i and the eoltsnn vectors are defined by Cl-au)u'-^ |<3). a-ap)w^ Again, note a chain of equalities: ptl) p«) p<3) -E«) P Dsing this ebain, we eenpute det C(c) • det (P -f c £) detp+c|^p'l' A...AP'«-" AE""j * and obtain t? d«t C(e)- A...AP<«-" AE"" e»o det ov)jH du K R(U> ]• where 5^^ > Kroneelcer delta.. Xo further sinplify this datezminant, we define the aatrices p(l) _ ^ _ fW) •'V RCu) e'" . . . s"" " = ( ifcr • Then det C(e)l - det [F - o G] •"e-O - - o A...A [F'"' - a G^^'j (A,4) 273 Next, we use a ebaia of equalities indexed by the pewer of u la the integrand, rCl) - ?«) . ^(1) - M (2) .K+2 p(K-l) (&.5) ?(N) u» « Un) .K+l ;(H) and consider the coefficient of in (A.4). The coefficients of 0° and are clear. Vtie coefficients of the which rcaain Bust be exterior products of j - "G-vectors" and H-j "F-vectors". By the chain of equalities (A.5), all such terms vemish (because the products have repeated vectors) except two. Thus, we find ^ detCCcl • det F + (-a)" det G JCBQ Finally, using the equalities (A.5) yields ^det C(c) det det CD. V[ aJ det M^'^^-i'-det j-1 e«o Y det - I det j-0 j-1 ' ' " A I a . det (+) j-1 iriileh Mtablishes (&.2}. vv /» F-.«^ULr& I. ^/=V, * . V *k Rew.a'rKs: (V^ a . (ii) Te^cTi ±» a. ^>\ee"t. 276 1>L S ^ 3 H ' ! 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