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T T.TV/T.T Dissertation vJ 1VJLJLInformation Service

University Microfilms International A Bell & Howell Information Com pany 300 N. Zeeb Road, Ann Arbor, Michigan 48106

8618804

Lee, Jong-Eao John

THE INVERSE SPECTRAL SOLUTION, MODULATION THEORY AND LINEARIZED STABILITY ANALYSIS OF N-PHASE, QUASI-PERIODIC SOLUTIONS OF THE NONLINEAR SCHRODINGER EQUATION

The Ohio State University Ph.D. 1986

University Microfilms International 300 N. Zeeb Road, Ann Arbor, Ml 48106

THE INVERSE SPECTRAL SOLUTION, MODULATION THEORY

AND LINEARIZED STABILITY ANALYSIS OF N-PHASE,

QUASI-PERIODIC SOLUTIONS OF THE NONLINEAR

SCHRODINGER EQUATION

DISSERTATION

Presented in P a rtia l F u lfillm e n t of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of the Ohio State University

Jong-Eao John Lee, B .S., M.S.

*****

The Ohio State University

1986

D issertation Committee: Approved by

Francis Carroll

Ted Scheick "^}1a Cv'CLflC^-l - Bostwick Wyman Advisfcr Department of Mathematics To My Parents

i i ACKNOWLEDGMENT

I t is my p riv ileg e to have Dr. Gregory Forest as my academic adviser. Without his enormous help, in both academic and personal aspects, I would not have been able to complete my thesis. I thank him with my greatest respect.

I also thank my graduate committee members, Dr. Francis Carroll,

Dr. Ted Scheick, Dr. Bostwick Wyman, for th e ir encouragement.

My wife Meili makes it possible for me to concentrate on my research through the years. I appreciate, with all my heart, her understanding and tremendous patience. My son Michael is an angel. He always inspires me to work harder. I also thank him.

I am very grateful to Mrs. England for her tremendous e ffo rts in typing this paper. VITA

A pril 6, 1953 ...... Born - Chaochou, Taiwan, Republic of China

1976 ...... B .S ., National Tsing-Hwa U niversity, Taiwan

1977 ...... Teaching Assistant, Department of S ta tis tic s , Catholic Fu-Jen U niversity, Taiwan

1978 ...... M .S., Western Illin o is U niversity, Macomb, Illin o is

1978-1981 ...... Teaching Assistant, Department of Mathematics, Michigan State University, East Lansing, Michigan

1981 ...... M.S., Michigan State University, East Lansing, Michigan

1981-present ...... Teaching Assistant, Department of Mathematics, The Ohio State U niversity, Columbus, Ohio

PUBLICATIONS

Forest, M. G ., and Lee, J. E ., Geometry and Modulation Theory for the Periodic Nonlinear Schrodinger Equation, Proc. IMA, Univ. Minn., Workshop on O s cillatio n Theory, Computation, and Methods of Compensated Compactness, 1985.

FIELD OF STUDY

Major Field: Nonlinear Partial Differential Equations

iv TABLE OF CONTENTS

DEDICATION i i

ACKNOWLEDGEMENTS i i i

VITA iv

LIST OF FIGURES v ii

ABSTRACT v iii

CHAPTER PAGE

I INTRODUCTION 1

I I THE PERIODIC NLS SPECTRAL THEORY 4 1. Floquet Theory 4 2. Floquet Theory and Scattering Theory 8 3. The Periodic NLS Spectrum 13 4. A Special Example 16

I I I THE N-PHASE NLS WAVETRAINS 19 1. 2N Simple Spectra; Determination of a Riemann Surface 19 2. Quadratic Eigenfunctions 20 3. Construction of Special Solutions; u-Representation of N-Phase NLS Wavetrains 25 4. Theta Function Representation of N-Phase NLS Wavetrains 31 5. NLS Constraints 38

IV MODULATIONS OF N-PHASE NLS WAVETRAINS 41 1. Averaged Generating Functions 41 2. An In variant Representation of Averaged Generating Functions 47 3. An Invariant Representation of the Modulation Equations; Consequences of the Invariant Representaton 49

v CHAPTER PAGE

V THE LINEARIZED INSTABILITIES OF N-PHASE 54 NLS WAVETRAINS 1. The Linearized Instability of the NLS Plane Wave Solution 54 2. y-Coordinates: A Geometric Realization of Linearized Instabilities 57 3. The Analytic Argument of Linearized Instability 60

VI CONCLUDING SUMMARY; FUTURE RESEARCH 68 1. Concluding Summary 68 2. Future Research 69

APPENDICES

A PROOF OF ( I I I . 22) 71

B PROOF OF ( I I I . 23, 24) 74

C PROOF OF ( I I I . 29, 30) 77

D PROOF OF ( I I I . 33) 80

E PROOF OF THEOREM I I I . 9 83

F PROOF OF ( IV .7, 8, 9) 91

G PROOF OF ( IV .13) 103

H PROOF OF ( IV .15) 105

I PROOF OF ( IV .18) 107

J PROOF OF (V.14) 114

K PROOF OF THEOREM V.3 119

L INFINITE-PRODUCT REPRESENTATION OF A2(E )-4 123

M CONNECTION BETWEEN DATE y . VARIABLES AND THE 126 "y SPECTRUM" J

LIST OF REFERENCES 131 vi LIST OF FIGURES

FIGURES PAGE

1 N=3-phase periodic NLS spectrum 16

2 Basis of y-cycles fo r N=3-phase NLS solution 16

3 The spectrum of NLS plane wave with spatial period L 18

4 Homology basis on the Riemann surface, N=3 32

5 The lin earized growth rate of NLS plane wave 55

6 A real double point s p littin g into simple spectra 58

7 A nonreal double point s p littin g into simple spectra 59

8 Geometric evidences of stability of the linearized x-flow and instability of the linearized t-flow 60

9 Integral paths to evaluate the complex conjugate of 80 theta function representation of a N-phase NLS 81 wavetrai n 84

10 Integral paths to evaluate the growth rates of 117 lin e a rize d unstable modes of N-phasd NLS 118 wavetrains

v ii ABSTRACT

We study the nonlinear Schrodinger (NLS) equation under periodic boundary conditions. F irs t we focus on the exact theory of the periodic and quasi periodic NLS solutions which can be expressed in terms of Riemann theta functions. Next, we study certain perturbations of these solutions; namely, the slow modulations in space and time, perhaps in the presence of external perturbations, and perturbation of

initial conditions (linearized instabilities). We use Floquet theory and scattering theory to investigate the periodic NLS spectrum of the

AKNS linear system and use the Date technique to develop the inverse spectral transform ation. We fin d exact representations, with a minimal set of parameters, for N-phase NLS wavetrains. We give precise formulas for the physical wavenumbers and frequencies of these solutions. We then use multiphase averaging, a method due to Whitham and Flaschka, Forest and McLaughlin, to derive the modulation equations for N-phase NLS wavetrains. The modulation equations are then diagonalized, with explicit formulas for Riemann invariants and their characteristic speeds. We follow the work of Ercolani, Forest and

McLaughlin on the sine-Gordon equation to approach the linearized problem fo r a ll N-phase NLS wavetrains, from both geometric and analytic viewpoints. We characterize the stable and unstable modes.

The growth rates are computed e x p lic itly .

v iii CHAPTER I

INTRODUCTION

The goal of this dissertation is a study of the exact and perturbative structure of quasiperiodic solutions of the nonlinear

Schrodinger (NLS) equation. Before beginning, we place this study in the context of previous developments of th is fie ld .

NLS is one of several soliton equations; others include

Korteweg-deVries (KdV), sine-Gordon (s-G), sinh-Gordon (sh-G) equations,

q* ■ 6qq + q = 0 (KdV) , (1 .1 ) t x xxx

u - u + sin u = 0 (s-G) , (1.2) t t xx

iq + q + 2 |q |2q = 0 (NLS) . (1 .3 ) w XX

These equations have been derived in many physical contexts, as the model equations for a specific balance of physical influences. KdV is a generic weakly nonlinear, dispersive wave equation, modelling long waves in shallow water [58, 67]. Sine-Gordon is a special field theory equation, consisting of the wave equation with a special choice

1 2

of nonlinearity. It has been derived in the dislocation theory of

crystals [67], in superconductivity [33] and the geometry of surfaces

of constant negative curvature. The NLS equation is a model envelope

equation, combining dispersion and weak nonlinearity. It has been

derived in many physical contexts and is perhaps the most physically

important of the solition equations. Some examples of applications of

NLS are in optical fiber transmissions, lasers [33, 76, 78, 79], and

deep water theory [67]. In particular, NLS is easily derived in a

slowly varying envelope approximation from KdV and s-G .

Since these equations (and NLS in particular) are generic model

equations, their solution spaces and perturbations of these solutions

are worthy of study. These equations are tru ly special, however, in

that exact representations of classes of solutions have been found.

These include the famous soli ton solutions [25, 26, 36, 38, 67, 73],

the rational solutions [77], and the theta function (N-phase) solutions

[41, 42, 43, 46, 59, 2-6, 51, 53]. Our focus is the N-phase wavetrains of NLS.

Previous results on the exact theta function solutions have appeared in Its and Kotlyarov [51], Ablowitz and Ma [52], Previato

[5 4 ], and Tracy [5 3 ]. We w ill only improve these results s lig h tly to the extent that we give the sharpest representation, with a sharp count of free parameters in these solutions, and precise formulas for the physical wavenumbers and frequencies of these wavetrains. 3

Modulational instabilities are well-known for NLS on the ~-iine

[58, 67]. Here, these long wavelength instabilities are known to

saturate by the formation of NLS soli tons. Tracy [53] has exhibited

these long wavelength in s ta b ilitie s for the NLS plane wave, and done

the analysis using the inverse spectral structure. We aim to carry out

th is lin e a r analysis for a rb itra ry N-phase NLS solutions.

For stability considerations of N-phase waves we require further

results on another equivalent representation of these exact N-phase

solutions, the so-called ^-representation. We follow Ercolani and

Forest [4] and Forest and McLaughlin [3] to establish the topological

structure of these dynamical coordinates for NLS isospectral sets.

These results are crucial in the modulation theory and linearized

stability analysis of NLS solutions, and are new.

Next, we proceed to study the perturbation theory of this class of

NLS solutions. The modulation theory is developed to the same level of mathematical structure as the previous studies for KdV [1] and

sine-Gordon and sinh-Gordon [3, 5]. The "linearized analysis is then carried out using the y-representation and the technique of multiphase averaging. All linearized instabilities are identified, and their growth rates are e x p lic itly derived. CHAPTER I I

THE PERIODIC NLS SPECTRAL THEORY

It is now well known that a soliton equation under vanishing

boundary conditions can be integrated by the inverse scattering

transformation (1ST) [38]. The vanishing boundary conditions

eventually determine the scattering data to recover the potential. For

the periodic boundary conditions, we need the inverse spectral

transformation. Unlike the periodic KdV equation, where the linear

Schrodinger operator is s e lfa d jo in t and the spectrum is easily

understood, the periodic s-G and NLS spectrum are much more

complicated. Forest and McLaughlin ([2,61]) analyzed the periodic s-G

spectrum in detail. We now follow [2,61] to develop the periodic NLS

spectrum. This method is based on Floquet theory and scattering

theory, which we now describe.

I I . l Floquet Theory

The NLS equation,

iqt + qxx + 2 h l 2q = 0 H l . l ) arises as the compatibility condition for the Z-S [36] or AKNS [38] system: 5

’ = + * ( II.2 a )

' = " ^ 1 + 1E^2 ' and

( * 1 >t = U q r - 2E 1^ + (iqx + 2EqH2 , ( II.2 b )

, (2) t = (1 r - 2Er)iJ»1 + i(2E - qrty .

where

r = q ( II.2 c )

( ) denotes complex conjugate.

A Straightforward Fact: if $(x,t;E) = (^(x.tjE),<|>2(x,t;E)) is a solution of (II.2a,c), so is $(x,t;E) where

$(x,t;E) = (<{>*(x,t;E*),-<(>*(x,t;E*))T ; (II.3)

moreover, $(x+L,t;E) is also a solution of (II.2a,c) provided that

q(x+L,t) = q(x,t) , -» < x < « . (II.4)

We are interested in the spatial periodic NLS solution q(x,t) ,

(11.4), which, of course, is an isospectral flow of the system

(II.2a,c). To investigate the periodic NLS spectrum, it is enough to consider the initial system of (II.2a,c), where

★ qQ(x) = q(x,0) , qQ(x + L) = qQ(x) and rQ(x) = qQ(x) . (II.5b)

We now fix x„ in the x-axis and consider the fundamental basis — j {$,$} for the system ( I I . 5) where $ = t ^ , ^ ) and

$(x=xq;E) = (1.0)T ; 7(x=xQ;E) = (0 ,-l)T . ( I I . 6)

From the above fact (11.3,4) and a straightforward calculation, we fin d

($(x+L,E),7(x+L;E))T = T(E)(*(x ;E),7(x;E))T ( II.7 a ) where the transfer matrix T(E) = [T. .1 „ is given by ------l j 2 x 2

Tn (E) = x( xq+LiE) , Ti 2(E) = -4»2(xo+L;E) ,

( II.7 b )

T2 i(E) = “Ti2(E*) » T22(E) = T*X(E*) .

The eigenvalues of T(E) , p+(E) , satisfy 7

p2(E) - A(E)p(E) + det T(E) = 0 (II.8a)

where

* * (i) A(E) = trace T(E) = P+(E) + p_(E) = TU (E) + )

★ ★ = 1 (X0+L; E ) , ( II.8 b )

( i i ) P+(E) . p (E) = det T(E) = 1 , ( II.8 c )

A(E) ± V?7 a (E) - 4 ( i i i ) P+ ( E ) ------T------( II.8 d )

The function a (E) is called the Floquet discriminant and is fundamental to the entire periodic NLS spectral theory, as indicated by

a = spectrum of(II.5)={E€C, -2j

which is derived from (1 1 .7 ,8 ) where -2 < a (E) < 2 i f f Ip (E )| = 1 — — ± iff the corresponding eigenfunctions of T(E) (which are solutions of

(11.5)) are bounded. Indeed, the NLS isospectrality is now captured by

A(E|q(x,t)) = 0 , q(x,t=0) = qQ(x) , (11.10)

which is verified by the scattering representation of a (E) (re fe r to

(1 1 .2 2 )). 8

We now investigate a CE) to realize the periodic NLS spectrum.

Lemma I I .1 (Basic properties of the Floquet Discriminant ME))

(i) ME*) = A*(E) . (11.11)

(ii) For real E , -2

These are straightforward results of (11.8).

I I . 2 Floquet Theory and Scattering Theory

The Floquet discriminant a (E) can be represented in terms of well known scattering data of (11.5a) for certain localized potential

qQ , from which we are able to analyze M E ) .

Consider a localized initial NLS potential q 0 ’

!qQ(x)| 0 as |x| near - , rQ(x) = qQ(x) . (11.13)

The Z-S system, (II.5a,13), as |x| near » , is deduced to

*i = -1E*i ■ (11.14)

i i m iE*2 ■

which admits the fundamental basis {y .v ) ,

T - 1E Mx;E) = (1,0) e (11.15) T iE (x - X0’ T(xiE) = (0,-1) e 9

For the exact system (II.5 a ,13), there are two independent bases,

{F,F} and { G,G} , known as the Jost functions, such that

F~y , F~'f as x near ~ , (11.16)

G ~ w , G ~ 'F as x near -°° .

Suppose th a t qQ has compact support, AKNS [38] showed {F,F,G,G} are entire functions of E , and

(F,F)T = M(E)(G,G)T (II.17a) where the tran s fe r matrix M(E) = [M ] is given by i j 2 x 2

Mu(E) = Wronskian(F,G) , M^(E) = Wronskian(G,F) ,

(II.1 7 b ) 'if M21 ( E) = -M12(E ) , M22(E) = Mn (E ) , and

det M = 1 . (II.17c)

Moreover, {M (E)} are entire functions and i j

M11(E),M22(E) ~ 1 '

as |E J near « . (11.18)

M (E),M (E) ~ 0 , 12 21 10

We now take qQ to be the truncated potential from the

L-periodic potential qQ , (II.5b),

A qQ(x) = qQ(x) , xQ < x _< xQ + L ,

(11.19) = 0 , otherwise.

Then the asymptotic arguments, (11.14, 16, 18), are now applied to x near and x^+L , corresponding to -» and » . By straightforward calculation, (11.17) now becomes

G(x=xq+L;E) = Mn (E)e lEL • G(x=xQ;E) + M12(E)eiEL • G(x=xQ;E) ,

( 11. 20)

G(x=xQ+LiE) = -M12(E )e" • G(x=xQiE) + Mn (E )e' • G(x=xQ;E) .

By observing the two systems, ( II.5) and (II.5a, 19), they are identical in the period xQ £ x jc x^+L ; moreover, from the asymptotic behaviors of {G,G} a t xQ , {G,G} is precisely the fundamental basis, (II.6), in this specified period. Consequently, the two transfer matrices, ( II.7) and (11.20), are identical, which immediately yield the following relations,

T11(E) = Mn (E>e"lEL » t 12(e> = M12(E)elEL , (11.21)

T21 (E) = "T12(E* } ' T22(E) = T11(E* } * 11

Lemma I I . 2 (Scattering Representation of the Floquet Discriminant a (E))

( i ) a (E) is an e n tire function which can be expressed in terms

of the scattering data:

_i EL * * iEL A(E) = Mn(E)e + Mn(E )e (II.22a)

where

★ ★ M11(E) * M11(E ) ~ 1 aS ,E* near " * (II.22b)

(ii) For real E ,

-2 < a(E) = 2|M (E)|cos(ph(M (E)) - EL) < 2 , (II.2 3 a ) where

|M (E)| <1 and M (E) = |M (E)|exp i(ph((M (E))) . (II.23b)

(iii) If E is real and a (E ) = ±2 , then a 1( E ) = 0 along K k K the real E-line. (11.24)

Remarks on the Proof: (i) (11.22) follows (II.8b, 21). {M..(E)} are J entire functions implies A(E) is also an entire function. The scattering representation of a (E) , (11.22), is most essential in understanding the periodic NLS spectrum. It determines the asymptotic behaviors of a (E) near |E| = ~ and shows 12

^ A (E |q (x ,t)) = 0 , (11.25)

whenever q evolves according to NLS , which is due to that ★ * M11(E|q(x,t)) , M (E |q(x,t)) are invariant of t (a well known fact in inverse scattering theory where rp- is the transmission 11 coefficient). Moreover, under the Hamiltonian structure, the periodic

spectrum, £(q(x,t)) = {E , a (E ) = ±2} , provides the action variables k k in action-angle coordinates (see [70, 61, 6]). (ii) For real E , det M(E) = |M (E )j^ + |M ^ (E )|2 = 1 , i . e . , (II.1 7 c ) , which implies

|M (E)| j< 1 . (iii) (11.24) is verified by straightforward ■k ★ calculation of a (E) = + ^ ( E ) in (H .8 b ) (also is proved in [5 2 ]).

Lemma I I . 3 (Asymptotic Behaviors of the Floquet Discriminant a (E)

near |E| = °°)

. , iEL -iEL o A ( E) — e + e = 2 cos (EL) (II.2 6 a ) = 2[cos(ReE«L)cosh(ImE«L) + i sin(ReE»L)sinh(ImE»L)] where

E = ReE + ilmE . (II.26b)

In particular,

( i ) a (E) ~ 2 cos(EL) when E is real and E ~ . (II.27a) 13

( i i ) a (E) ~ 2 cosh(|E|L) when E is purely imaginary and

|Ej ~ ~ . (II.27b)

( i i i ) |a (E)J -»■ * when JImEj + «o . (II.2 7 c )

( iv ) |a ( E) | is bounded and a (E) oscillates as (11.26) when

| ImE|

(v) |A(E)| > 2 when a (E) is real, E is nonreal and

| E j ~ oo . ( I I . 27e)

Remark: these results are straightforward from the scattering representation of a (E) , (11.22).

I I . 3 The Periodic NLS Spectrum

Theorem I I . 4 (The periodic NLS Spectrum)

Assume that q(x,t) is a solution of NLS , (II.l), with spatial period L . Then

a (q ) = spectrum ( II.2 a ,c ) = {E € C , A( Ejq) 6 R , -2 < a (E) £ 2} .

(II.2 8 a )

In particular, if

z(q) = {E € a ( q) , a (E ) = ±2} , the periodic spectrum, (II.28b) K K

s z (q) = {E^ £ z(q) , A1(E^) * 0} , the simple spectrum, (II.2 8 c ) 14

and

z d(q) = {E 6 z(q ) , a '(E ) = 0} the double spectrum, (II.2 8 d ) IV I v

s d then (i) a(q) cjjq ) = e (q) U z (q) . Each of these sets, say Q , s a tis fie s the symmetry:

E € Q i f f E* € Q . (11.29)

( i i ) The real lin e R is a continuous spectrum and has no simple point, i.e.,

R c a(q) and ZS(q) fl R = 0 . (11.30)

(iii) Ed(q) has at most finite elements in C \ R . s (iv ) i (q) has exactly N distinct complex conjugate pairs if q is an N-pahse NLS wavetrain which is 2ir-periodic in each of the phases (r e fe r to (1 1 1 .3 9 )).

(v) There is a one to one correspondence between degrees of freedom in q and critical points of a (E)q) . Each double point

E , € Ed labels a closed (unexcited) degree of freedom, while each d complex conjugate pair of z (q) corresponds to an open (excited) degree of freedom. For generic q ,

£d(q) = 0 (11.31) 15

Moreover, when a closed degree of freedom is excited, the corresponding double point E^ splits into a pair of simple spectra.

(vi) The function theory of q takes place on the Riemann surface ft of * ^ 2 (E|q) - 4 , with branch points E. precisely at s ^ the 2M, N _< oo , elements of z (q) . If N is finite, then the appropriate curve is

2N ^ R (E) = n (E - E.) , E„, = E9, . (11.32) L J 2 j - l 2j

Remarks on the Proof: (11.29, 30) are straightforward results of Lemma

I I . 1, I I . 2. Property ( i i i ) is proved in [7 ]. This may also be argued from the asymptotic behaviors of A(E) , (II.27e). This property

(iii) places a bound on the number of linearized instabilities for given NLS solutions in Chapter V. Properties (iv,vi) are verified in Chapter I I I fo r the N-pahse NLS wavetrains. Property (v) is essential in multiphase averaging technique and perturbation argument in Chapters IV, V, and is proved in Chapters III, V (also see [2, 6] for the complete description of s-G problem). For finite N , the

NLS spectrum is illu s tra te d in Figure 1. s Remark: in addition to the simple spectrum, z (q) , we will investigate, in Chapter III, the Dirichlet eigenvalues, {u.} , of the J Z-S eigenvalue problem (II.2a,c) to carry out the inverse spectral transformation for N-phase NLS wavetrains and s ta b ility analysis of these wavetrains. The y-variables are not isospectral and behave very complicated (refer to (I11.23)), yet there are simple homologous curves for p-variables, as in Figure 2. 16

N=3-phase NLS spectrum Basis of \i-variables for N=3-phase NLS Solution

Fig. 1 Fig. 2

I I . 4 A Special Example

Consider the NLS plane wave solution q(x,t) ,

. 2 q(x,t) = aeia , (11.33)

where a is real, and let L be the spatial period of q . The Z-S eigenvalue problem ( I I . 5) now is easy to integrate (dQ(x) = a , a constant). In particular, the fundamental basis is given by

E-a ia .T ixx ,E+a - i a NT -ix x (x;E) = ( ) e + (: (II.3 4 a ) •2x ’ 2a •2a 2A

-Aa E-a.T -iAx , ,Aa E+A,T iAx

where

-V777. (II.3 4 c )

From ( II.8), we find

+ 2 L ,(E) = 2 cos('V VEe2 + + a2 a . • LI L) ,A( , (11.35)

. 2 and the periodic NLS spectrum, o(q = ae ) , satisfies

( i ) Ek € E i f Ek = ±1 :L- r - - a2 , (11.36) - n -

in which case, E is real or purely imaginary. In particular, z k has only finite purely imaginary elements.

(ii) z = {±ia} = {E0,Eq} . (11.37)

2 / d (iii) If L < — , z has no purely imaginary element; if

2 2 3 2 2 n ir ,2 (n+1) V d u . , kl — — < L < ------, then z has precisely N pairs a a of complex conjugates in purely imaginary line.

The periodic NLS spectrum is depicted in Figure 3. We will return to this example in the analysis of linearized instabilities in Chapter V. 18

:: E„ = i I a I

-•O’ —^-<3 -

■i|a|

ia t The spectrum of NLS plane wave q = ae with spatial Period L .

* denotes E € ES = {±ia} j ------d Z k V 2 o denotes E 6 e = { / — - a , k * 0} d V L2

Fig. 3 CHAPTER I I I

THE N-PHASE NLS WAVETRAINS

For quasi periodic solutions of soli ton equations, there is a

straightforward construction, due to Date [59]. Ercolani, Forest and

McLaughlin [2-4] have carried out this analysis for s-G. We now

follow [59], [2-4] to develop the NLS inverse spectral s transformation. In particular, if the simple spectrum z has finite

(2N) elements, we find the inverse spectral solutions have precisely N

degrees of freedom and are N-phase wavetrains, where N-l degrees of

freedom are parameterized by N-l p-variables (Dirichlet eigenvalues

of Z-S (II.2 a )) and the remaining one corresponds to the plane wave

factor attached to the theta function representation of those N-phase

NLS wavetrains (s e e (III.39)).

I I I . l 2N Simple Spectra; Determination of a Riemann Surface

Given a prescribed simple spectrum zS , we now construct a

Riemann surface where the function theory of the corresponding periodic

NLS solutions take place. The eigenvalues ( I I . 8) of Z-S eigenvalue problem ( I I . 5) indicate that the proper Riemann surface ft is given

A by the function R(E) ,

R(E) = ( a2(E) - 4)2 , ( I I I . l a )

19 which has branch points precisely at the simple spectrum £S . In s 2N particular, if e has only 2N (fin ite ) elements, with ★•i. - K x E2 k l = Egk , then R(E) can be written (Appendix L, also see [2,61] for s-G) as

R(E) = S(E) . ( n (E - Ek) ) 2 , ( I I I . l b )

where S(E) is a single-valued function. We focus on the hyperelliptic curve, (E,R(E)) , where

2 2N R (E) = n (E - E, ) . ( I I I . 2 1 k

We now pose the fin ite degree of freedom inverse problem: given 2N N distinct pairs of complex conjugates (E .}. , determine the most 2N general periodic NLS solutions with simple spectrum {E } . This l \ X can be achieved by the inverse spectral transformation we now develop.

I I I . 2 Quadratic Eigenfunctions

We now employ an equivalent system of quadratic eigenfunctions

{f,g,h} for the Z-S eigenvalue problem ( I I . 2),

h = ^ , ( I I I . 3)

where $ = and v = are solutions of ( I I . 2). By straightforward calculation, we find 21

r f x = "r9 + qh *

' 9X = 2qf -2iEg , (III.4 a )

, h = -2 rf + 2iEh , ' x and

f = i ( r - 2Er)g + i(q + 2Eq)h , t x x

< g = 2(iq + 2Eq)f + 2i(qr - 2E2)g , (III.4 b ) v X.

h = 2(ir - 2Er)f + 2i(qr - 2E2)h , D X where

* r = q . (III.4 c ) It is easy to verify that

(f,9,h)xt = (f,9,h)tx iff iqt + qxx + 2|q,2q = ° (111.5) * and r = q .

The quadraticeigenfunction system ( I I 1.4) isfundamental to the theory of the NLS equation, as indicated in

Lemma I I I . l (Basic Properties of Quadratic Eigenfunctions)

Suppose that {f,g,h} is a solution of (III.4), then

2 3 P 3 P (i) P(E) = f - gh is a first integral, i.e., — = — = 0 .

(111.6) 22

( i i ) g,h satisfy the linearized NLS equation about an arbitrary solution q :

2 igt + 9xx = 2q h " 4qrg * 2 r = q* . (III.7) ih,. - h = 4qr - 2r g , t xx

( i i i ) {f,g,h} satisfies the fundamental conservation law:

f - [4Ef + i(qh + rg)] = 0 . (III.8) t x

These results are consequences of straightforward calculations from

(III.4, 5).

Some Remarks: (i ) The fir s t integral P(E) introduces the hyperelliptic curve (I I I .2) in the context of the periodic NLS solutions with N degrees of freedom in Sec 3. ( i i ) Clearly, the quadratic eigenfunctions {f,g,h} w ill play a fundamental role in local perturbation analysis of the NLS equation. We analyze linearized instabilities of the NLS solutions in Chapter V.

( i i i ) The fundamental conservation law ( I I I . 8) actually provides the entire in fin ite family of NLS conservation laws. Moreover, this representation is central to the modulation theory of the N-phase

NLS wavetrains in Chapter IV.

We now normalize the linear system ( I I I . 4) by the normalization condition:

f2 - gh = 1 , ( I I I . 9) 23

which, from ( I I I . 4), implies that

f ~ 1 + I f (x,t)E~J , 3=2 3

9 ~ I g .(x,t)E J , as E near ® . ( I I I . 10) • 1 J J=1

h ~ I h (x,t)E'J , 3=1 3

Inserting ( I I I . 10) into ( I I I . 4) yields the following equations for the coefficients {f.,g.,h.} : 3 3 3

g: = -iq , hx = -ir ,

(III.1 1 a ) (f.) = -rg. + qh. , J x j j

{Vx= 2qfj - 21 Vi •

(h ) = -2rf + 2ih , j > 1 , 3 x 3 3+1 and

( f .) . = ir g. - 2rg + iq h. + 2qh , 3 t xa3 aj+ l "x j 3+1

(9A = 2i9¥f, + 49f•., + 2iqrg - 4ig , (III.lib) 3 t X 3 3+1 3 3+2

(h.) = 2ir f + 4rf - 2iqrh + 4ih , j > 1 . 3 t x 3 3+1 J j+2 - 24

From the normalization condition (III.9 ), it is straightforward that

one can generate recurrsion formulae for { f.,g .,h .} in terms of the * J J J potential q, r = q and their spatial derivatives. For convenience,

we lis t the fir s t few:

fo = 1 * 9i = "iq * hi = "ir ' fl = ° *

V - I qr> 92=7 \ ’ h2 = -2 rx-

f3 = 4 (qrx ' ’ 93 = 2 rq2+i qxx ’ h3 = I ^ + i r xx ’

3 2 2 3 1 , , ( U I - 12> V s qr - iV x +I (rqx + qrx>x ’

3 1 3 1 g = - — rqq - — q , h = — rqr + — r 4 4 x 8 xxx 4 4 x 8 xxx

We now insert (111.10) into the fundamental conservation law ( I I I . 8 ) , which yields the in fin ite family of conservation laws:

I {(f.) - [4f. + iqh. + rg.)] }E J = 0 as E near ~ , (III.13a) j t j+i j j x J

i . e . ,

(f.) - [4f. + i(qh. + rg.)] = 0 , j>2. (III.1 3 b ) J t j+1 J J x

From ( I I I . 12), the fir s t three conservation integrals corresponding to

( I I I . 13) are: which correspond to the number of particles, the momentum and energy respectively. In general,

L V 2 ci_? = I | Mx,t)dx , j _> 2 . (III.15) J ” y -L J O ' 2

I I I . 3 Construction of Special Solutions; ^-Representation of N-Phase Wavetrains

We now restrict a periodic NLS spectrum with exactly N 2N distinct complex conjugate pairs of simple spectrum {E } . Date

s 2N [59] showed that N-phase NLS solutions have z = ^ and these potentials q correspond to a polynomial ansatz on quadratic eigenfunctions up to a constant factor. Date Ansatz: assume that

N f = I f (x,t)E3 , j=0 3 N-l N-l g = I g (x,t)E3 = g n (E - y ) , ( I I I . 16) j=0 3 1 N-l . N-l h = I h.(x,t)E3 = h n (E - p.) . j=0 3 l 3

Lemma I I I . 2 (Existence of Polynomial Quadratic Eigenfunctions)

Polynomial solutions ( I I I . 16) of the quadratic eigenfunction system ( II 1 .4) exist i f f the NLS potential q has N degrees of 2N freedom. Moreover, Un q) can be represented in terms of {E > X K J. t and the N-l zeros of g .

To prove this Lemma, we investigate the quadratic eigenfunctions

{f,g,h} in (III.16). Inserting (III.16) into (III.4) yields the following equations for coefficients { f.,g .,h .} : J J J 27

and

( f . ) = ir g. - 2rg.+ iq h. + 2qh. , , J t x j j-1 x j j-1

(g.) = 2iq f. + 4qf. + 2iqrg. - 4ig. , (III.17b) J t x j j-1 j j-2

(h.) = 2ir f. - 4rf. - 2iqrh. - 4ih. „ . j t x j j-1 j j-2

2N 2 The connection between {E } and the fir s t integral f -gh with the Iv X choice ( I I I . 16) is

2 2N f - gh = n (E - E ) . ( I I I . 18) 1 k

We now normalize {f,g,h} by Yf - gh = M n (E - Ek) to v i 2 connect the normalization condition f - gh = 1 , (III.9). This makes sure that the corresponding conservation laws are the exact physical ones and generates the recurrsion formulae for {f ,g ,h } , j j j where the fir s t few coefficients for the ansatz ( I I I . 16) are:

i 2N fN - 1 • Vl = <- 21 I Ek ’ %-! = -1" • Vl = -''r 1

1 2N ! 2N 2 fN-2 2 t. ^ . EiEj ' ql' ‘ 4 1 J Ek’ 1 ’ 1 > J 1 (III.19a) 1 i 2N 9N-2 = 2 qx + 2 \ Ek^ ■ 2N

\-2 = * I r* + 2 V • 28

From the fundamental conservation law ( I I I . 8), the generating functions

f and 4Ef + i(qh + rg) satisfy

N 2N ( i) f = I f E3 where f = 1 , f = (- - ) £ E . 0 3 1 (III.19b) N+l 2N ( i i ) 4Ef+i(qh+rg) = £ d.E3 where d..x1 = 4 , d = -2 £ E , q J N+l N “ k

1 2N 2 V l = 2 Ei Ej " 2^ I V ’ i>j 1

We notice that {f..,f., . ,dkl ^d^.d^, are constants, independent of N N-l N+l N N-l x and t . This fact becomes essential when we investigate modulations of N-phase NLS solutions in Chapter IV.

From (II1 .1 9 a ), we find the preliminary inversion formula:

' \ + Ek)" ■ v 2 > ( I I I . 20)

3 2 2N i qt + 21[I ‘S Ek> - l - 2( \ EK)9N-2 + % - 3 • j>k 1

From ( I I I . 16) and (III.1 9 a ), g can be represented as

N -l g = (-iq) n (E - u.) , (III.21) 1 3 N -l where the zeros { y .} of g turn out of the Dirichlet eigenvalues J i of the Z-S system (II.2 a ). We defer the proof in Appendix M, a method due to Forest and McLaughlin [61]. From ( I I 1 . 16, 18, 21), we summarize a series of facts and leave the proofs in Appendices A and B. 29

Lemma I I I . 3 (Properties of Polynomial Quadratic Eigenfunctions)

Assume that {f,g ,h } ( I I I . 16) is a solution of the quadratic eigenfunction system ( I I I . 4 ), and the normalization condition ( I I I . 18)

holds. Then

2N } (i) f. €R,0

(ii) h. = -g* , 0 < j < N-l . (III.22b) J J

* * N-1 * (iii) h(x,t;E) = -g (x,t;E ) = (-ir) n (E - p.(x,t)) ,

* (111.22c) r = q

(iv ) (Generating Functions in terms of y-Variables)

2N N-l N-l f -

„ N-l „ N-l 2 2 * rg = -i|q | n (E - y ) , qh = -i|q | n (E - y ) , 1 J i J where

2 i \ 1 \x . qxx ,V

I "I * - 2 V ' 2 ~ Wlth C^ }* + C7 ]' 30

Lemma 111.4 (Dynamical System for the u-Variables)

N -l Under the assumption of Lemma 111.3, satisfy

R(y )

V * = (-2,)^ - ( I H -23a)

" W j*k , 1 < k £ N -l .

2N N -l

V t = Vx ' ( 1 E£ " 2 1 (H I.23b ) £=1 j*k

Lemma I I I . 5 ((an q) has N-l Degrees of Freedom) _ — X t

Under the assumption of Lemma I I I . 3, the NLS potential q satisfies

q N -l 2N r m iV l vi -J Ek^ d ll.2 4 a ) 1 1

q ~ 2n

q j>k J K l K (III.2 4 b ) 2N N-l - 4 i[(- - I E ) .( I + I Vil 1 1 J i > j

Some Remarks: ( I I I . 22) is important in developing the modulation theory in Chapter IV where, in particular, the averaging technique 31

heavily depends on the y-symmetric structure of the generating functions (III.22d); moreover, the structure of {f.,g.,h.} J J J (III.2 2 a ,b ,c ), eventually, assures the modulational instability of

N-phase NLS solutions. (III.23) indicates that y-variables have complicated dynamical behaviors, just like s-G [2 ], yet, as shown in N-l [4 ], {y .}, are homologous to N-l simple closed curves on the J 1 Riemann surface ffi(E) , ( I I I . 2), (refer to sec. 4) from which we carry out the stability argument of the perturbed NLS solutions. ( I I 1.24) is referred as the y-representation for the NLS solution q . This y-representation shows that Un q) has N-l degrees of freedom X

* N-l since the complete (x ,t) dependence is determined by {y.> . We J ^ w ill show that q has exactly N degrees of freedom where the remaining mode corresponds to the plane wave factor attached to the theta function representation of those N-phase wavetrains (see

( I I I . 39)).

I I I . 4 Theta Function Representation of N-Phase NLS Wavetrains

We now perform the Abel-Jacobi transformation [68] to linearize the y o.d.e's and develop the inverse spectral transformation to 2N integrate the NLS potential q with 2N simple spectra (E } . I v A

To define the Abel-Jacobi transformation, we introduce "canonical a-b cycles" on the Riemann surface R of R(E) , 32

R2(E) = n (E - E ) . ( I I I . 25) 1 k

In the special case, N = 3 , with the branch cuts given by the NLS

spectrum as in Figure 1 , a-cycles and b-cycles are depicted in

Figure 4,

upper sheet lower sheet

Homology basis on B , N = 3

Fig. 4

We now introduce N-l Abelian differentials of the fir s t kind,

N-l At, r _N-l-v dE . ^ . j = ix J' R(eT ’ 1 - J - " - 1 ■ where the constant matrix C = [C ..] is given by the normalization i j condition,

/ dli = 5 , 6 is the Kronecker symbol, 1 < i , j < N-l ( I I I . 27) a j ij ij - ~

Next, we define the period matrix B = [B ] , i j 33

B .. = / dU. , 1 < i

I t is clear that the matrices B,C are completely determined by 2N { E.} , independent of q (x ,t) . J ^

Lemma I I I . 6 (Properties of Constant Matrix C and Period Matrix B)

( i) C..€iR, 1 i,j _< N-l . (III.29) J

( i i ) B..€iR, Re(B. .) = - • — for i*j, ( I I I . 30) JJ IJ 2 1 < i,j £ N-l .

We defer the proof in Appendix C, which is based on the methods developed by Ercolani, Forest and McLaughlin [2-4], and Date [59].

We now define the Abel-Jacobi map from p to a where

J N-l , u — 6 R > (III.31a) 1 N - l and

£ = )T € CN“1/{e.,B., 1 < j < N-l} , (III.31b) 1 N-l j j — —

N-l -► where e. denotes the orthonormal vectors in R and B. is the J J j-th column of the period matrix B :

N-i p £.(p) = I / dU. , 1 _< j _< N-l . ( I I I . 32)

° k=1 v.k ° 34

Using the Lagrange Interpolation formulae (see [52]), we show, in

Appendix D, that {£.} are linear in both x and t , i.e ., 3

Lemma I I I . 7 (&-Variables are linear in x and t)

(i) (£.) = (-2i)C , (III.33a) J x Jl 2N (ii) U.) = (-2i)[( I E )C + 2C ] , (III.33b) J t ^ K JI

1 1 J 1 N_1 ‘ 2N (iii) ^(x.t) = (-2i){Cjxx + [( I + 2Cj2 ]t) + °z. , (III.34a)

(iv) s.. (x ,t) -2. € R , (III.34b) 3 3

As indicated in (III.24), for the NLS potential q , (nn q) is X t 2N N-l completely determined by { E } and { y .} . To find an explicit K X J 1 formula for u (x ,t) , where the (x ,t) dependence is given by the Abel-Jacobi map ( I I 1.32) restricted to the NLS flow ( I I I . 34), we need to solve the so called "Abel-Jacobi inversion problem" [68], whose solution is in terms of Riemann-theta functions [69]. This NLS inverse spectral solution has been formally derived ([51, 53, 66]), which provides the theta function representation for the NLS potential q , in the modified form ( I I I . 39), yet two fundemental questions were s till remaining:

(1) What are the real, 2Tr-periodic phases of q (x,t) : 35

e ) , 1 < j < N N — —

( I I I . 35)

(2) What is the N-phase isospectral class of {E.} the set

of all NLS potentials satisfying

★ 9 * * * 9 9***9 ( I I I . 36)

Remark: Ercolani, Forest, and McLaughlin [6] have shown that the

complete isospectral sets for non-selfadjoint theories (in particular,

s-G) may include separatrices. The same is true for NLS . The theta

function solutions do not capture these components, which are related

to complex double periodic spectra. The analogous theory for NLS

w ill appear in another place.

The above two fundamental questions ( I I 1.35, 36) w ill be answered

(for the fir s t time) through the remaining chapter. We now quote and modify the theta function representation of the NLS solution q

given by Its and Kotlyarov [51], Matveev [66], and Tracy [53]. First, we define the Riemann theta functon, ©(z|B) , for N-l variables,

©(z ]B) = I exp{Tri[ + 2]} . ( I I I . 37)

The "periodicity" properties are [69]:

(i) 0(z + e.|B) = g(z )B) , (III.3 8 a ) J

(ii) 0(z + B. IB) = [exp(-2iriz. -iriB..)] • © {z | B) , (III.38b) J 1 J JJ N-l N-l N-l ( i i i ) 0 (z+ I n e + I m B ) = [exp{-ui[ I m.B. .+2]}] - K K - K K i J JJ

x e(z|B) , (III.38c) where m ,n EZ , 1 < k < N-l . k k - -

Theorem 111.7 (Theta Function Representation of N-phase NLS Wavetrain)

A quasi periodic NLS solution q which yields the 2N simple 2N spectrum { E } is an N-phase wavetrain, K X

0 ( eN) = q(x=0,t=0) •e•e

0(1) 0(— e(x,t)+£+3) (III.3 9 a ) O where 6,£,d are (N-l)-vectors and e.. is scalar such that N

(i) e(x,t) = (e ,..., 6.. , )T i e .(x,t) = k.x + u.t 1 N-l j J J 1 < j < N (III.3 9 b ) where

2N k. = (—4xr i )C J ji • "j = ('4,,i)[< \ Ek)Cj l +2Cj2 ] •

1 < j < N-l (111.39c) and k ,u are certain constants (refer to (IV.15c,d)). N N 37

N-l _ N-l I ( i i ) A.=-B..- I / ( / K dU.JdU. + V / dU. 1 < j < N-l . J 2 jj l k ^ 1 J (III.3 9 e )

(iii) d. = / dU. , 1 < j <_ N-l . (III.39f) J J

Remarks: (1) Kotlyarov and Its [51] have shown that kN»“N are real constants, independent of x and t . Previato [54] provided rather simple forms for k and w (see (IV.15c,d)). (2) From Abel-Jacobi N N map ( I I I . 32, 33, 34),

£ .(x ,t) = t e.(x,t) + £. , 1 < j < N-l (III.40a) J ct\ j j — — and

e. , k. , oj. € R , 1 < j < N-l . (III.40b) J J J ~ ~

(3) The theta function representation of q , (III.3 9 a ), immediately shows that q is 2Tr-periodic in each real phase e. , 1 <_ j <_ N , J

q(e ,. . . , 0 +2ir 0 ) = q(0 , . . . , 0 0 ) ( I I I . 41) 1 J N 1 j N

This completely answers the fir s t fundamental question, ( I I I . 35).

We notice that the representaton (III.3 9 a ) of q with 2N simple spectrum shows q has N degrees of freedom, which are represented by N N real phases {e .K . On the other hand, from Abel-Jacobi inverse

-1 + + 1 N_1 ° map: V : z — ► p with i = — V e.e. + £ , we find j= i J 0 38

-1 ->■ •* ¥ U) = y , (III.42a)

-1 v (e.) = a.-cycle , 1 £ j £ N-l . (III.42b) J J

Consequently, p-variables are homologous to a-cycles [4 ]. This property plays the fundamental role in our modulation and linearized

in stab ility analysis (refer to Chapters IV, V). The Abel-Jacobi N-l inverse map also shows that q is parameterized by {p.}, and e., . j 1 N

I I I . 5 NLS Constraints

We recall the two fundamental questions ( I 11.35) and ( I I I . 36).

The real phases are computed in Sec. 4. We now derive the inverse ★ spectral representation of r(x,t) = q (x,t) . This theta function representation of r(x,t) can be established through the inverse spectral transformation as for the theta function representation of q •k » except that {p.} are now replaced by {p.} , and {£.} by {£.} J J J J where

* *.(!>= I J * dU. , 1 < j < N-l . (III.43) J p-i ° J - - k- l Hk

The above argument is based on (III.2 2 c ), which shows that Un r)

* can be represented in terms of {p.} just as Un q) in terms of j x t {p.} . Consequently, using the same argument as for q , we find r J N also can be represented in terms of samephases {0 .}. in J • 39

* Theorem I I I . 8 (Theta Function Representation of r(x ,t) = q (x ,t)) ★ Under the assumption of Theorem I I I . 7, r(x ,t) = q (x ,t) has the inverse spectral representation,

o 1 + ° + e(—- e(x,t)+g-£) -ie (x,t) /. . v , ~ s. ©(3-[Jl+d]) 2tt N rte, eM) = r{x=0,t=0) — • e I N o o 0(3 - £) 0(|^ 0(x,t)+6-[£+d]) (III.4 4 a )

v ■f where e , £ , d and e., are given in Theorem I I I . 7, and the N (N-l)-vector 3 is

N-l Sk 0• = I / * du- » 1 < j < N-l . (III.44b) J k = ! Sk J

Now, we evaluate t , 3 and 3 in Appendix E with results in

Theorem I I I . 9

o Let £ , d and 3 be (N-l)-vectors as given in (III.39e, f) and

(III.44b). Then 40

- ► l (i) d = — + ilmd mod a , (III.45a)

+ n ^ 1 , (ii) 6 6 R mod A , (III.4 5 b ) n -l N-l A = ^ I Vk + ^ mA,nk’mk f Z} * o (ill) Red - p) = — or 0 , 1 < j < N-l . (III.4 5 c )

We use these quantities to evaluate the complex conjugate of the theta

function representation (III.39) of q and find which is exactly

identical to the theta function representation of r , (111.44),

* r(e ,...,e ) = q (e ,...,e ) , (III.46) IN IN and, fin a lly , the question ( I I I . 36) is answered. CHAPTER IV

MODULATIONS OF N-PHASE NLS WAVETRAINS

We now consider an NLS wavetrain, * * * »eN»E^ E2^ ’ which appears locally as an exact N-phase solution (see ( I I I . 39)), but 2N whose parameters, « vary on large scales in both x and t :

E.(X,T), X = ex , T = et , 0 < e « 1 . We call this object a J modulating N-phase NLS wavetrain. We do not prove the existence of

such a solution to NLS , rather we assume i t (see McLaughlin [62]

for such questions). We now follow the methods of Ercolani, Flaschka,

Forest and McLaughlin [1, 3, 5] for KdV , sinh-Gordon and s-G to

approach the NLS modulations. Our results show that the structure of

the modulation equations of q.Je, e .E ,, . . . ,En ) is completely N 1 N 1 2N understood and the modulations are unstable for all NLS wavetrains to which this theory applies. We remark that this formalism, originally

due to Whitham [58], only applies to long wavelength solutions.

IV .1 Averaged Generating Functions

We consider a conservation law, 3 + Z = 0 , for an arbitrary t x NLS solution. We now evaluate 3 + Z on a modulating N-phase t x NLS wave, q ^ fe 0 n»E1»***»E2n^ *

41 42 h311 v+h3:1'V= uj nr*e ir>*1< v+(ko ^ +e :* 11 v • j j (IV .1) and demand that the phase average of this expression vanish. This yields, upon exploiting the 2Tr-periodicity in each phase e. , the J averaged conservation law:

8 3 — < S' (q )> + — = 0 , (IV.2a) 3T MN 3X V * ' 1 where the phase average <•> of any function F of q., is defined by N

^ ! 0 " • C d6l • " deNF(qN(9l ..... VS .....V ’ ’ (2ir ) (IV.2b) 2N and is computed for frozen values of {E (X,T)} . We notice that the

NLS densities S', and fluxes X. are independent of the plane wave J J phase e.. , as can be inspected from ( I I I . 12, 13, 22d). Thus (IV.2b) N collapses to an (N -l)-fo ld integral in the calculations to follow. I t is, therefore, that the averaged density <3> and the averaged flux 2N depend only on {E^fX.T)^ . To obtain a closed system of modulation equations, we must average 2N independent conservation laws.

Recall the fundamental conservation laws ( I I I . 8),

f - [4Ef + i(qh + rg)] = 0 , (IV.3) t x MSScsrsu*.

which generates all conservation laws ( I I I . 13) under the normalization

condition ( I I I . 9) as E near <*> . We want to average these terms,

thereby averaging the entire family of conservation laws. We normalize

the polynomial quadratic eigenfunctions {f,g,h} of q (e;E, , . . . , E 0 ) N 1 2N by

f - gh = R(E) = / n (E - E, ) , (IV .4)

then density and flux terms are derived from (III.2 2 d ) in the

u-coordinates, as given in

Lemma IV .1 (Averaged Generating Functions in terms of y-Coordinates) 2N N-l N-l N-l (i) = {(E- — I Ek)< n (E-ik)>+}« yyyy .

(IV.5a)

( i i ) = ,

2N 2N N-l i = {[2 £ EE - - ( £ E ) ]•< n (E-u.)> - 2 i > j 1 1 3

2N N-l N-l N-l N-l • I Ei< 1 n (E-u.)> + 4<[ I P.U.+ I (u.) ] 111 J i>j 1

2N n (E-y j)>} -j^ y . - (IV.5b) 44

(ill) <4Ef + i(qh + rg)>

2N 2N 2N = { [4E - 2( I E ) • E + 2 I E E ( J E )2] 1 1 J i 1 K

N-l 2N N-l N-l • < n (E — |i.)> + [4E - 2 X E ] • < ^ y. • H (E - y.)> 1 J 1 j= l J 1 3

N-l N-l N-l

+ 4<[ I y u. + I (y.) ] • n (E - y . )>} • (IV.5c) i> j J j= l J 1 J R(E)

Next, we compute (IV .5) by changing variables of integration from e to y . From ( I I I . 32, 33, 34, 40), the Abel-Jacobi map induces the Jacobian,

i4 i ■ • (iv-6' 3y k i

The product structure of (IV .5) and (IV .6), together with the u-homology result ( I I I . 42), allows the (N -l)-fo ld integrals to be factored into products of one-dimensional integrals [5]. We defer this rather long and nontrivial calculation to Appendix F with results in

Lemma IV .2 (Iterated Integral Representation of Averaged Generating

Functions)

N+1

m = 1 jo Dj' 1 ' , I V -7a) where 45

(-) (-) (-) 1 2N DN+1 = 0 ’ °N = 1 ’ V l = - I ( ^ k), ( IV. 7b) and

l \ Act N—j—1jN) 2N (N-j-l,N-l) D(-) _ det M 1 . det M j det M ‘ 2 1 j k 1 ’ det M

(IV.7c) 0 £ j £ N-2 .

N+l . . . (ii) <4Ef + i(qh + rg)> = { ID EJ} —— , (IV.8a) j=0 J RlE) where

( ) 2N 2N pNIi ■ 4 ■ v - - 2( ^Ek» • V i ■ ( 2 ^ Ei Er 2 C£Ek3 3 • (IV -8bl 1 i > j 1 and

t+\ MCN-l-j,N+l) 2N (N-j-l,N) D. = (-4 ^ ------+ 2( y E ) • ^ L S ------j 1 det M 1 j k' det M

♦ [f ( I Ek)2 - 2 I EjE ] • ^ M-detM- } . (IV.8c) 1 1 >J

0 <_ j <_ N-2 .

(±) ( i i i ) D. € R , 0 _< j £N+l . (IV .9) J

Here, M = C where C is the constant matrix given in ( I I I . 26, 27) with 46

Mij f * ' 1 HIT • «»•">•>

(j>k) and M ’ is identical to M except that the j-th column is

k dE , k dE , k dE ,T V Mi) • V Ml) ■ - • I* ,E M i)’ ' (IV-10b) 1 2 N-l

(±) (±) (±) Some Remarks: (1) D.. , , D., , D , , turn out to be the ------N+l N N-l coefficients of the exact generating functions (III.19b), which is essential in understanding the structure of 2N modulation equations 2N of {E (X,T)} as we describe later. (2) The real values of (±) k 1 {D. } , ( IV .9 ), are crucial in determining the stabilities of these 0 modulation equations.

Inserting (IV .7, 8) into (IV .2), we find the fir s t order NLS

N-phase modulation equations, a result of multiphase averaging technique,

N+i j N+l j

FT I S dj mu)+T x 4 V i W ' 0 ' ( I V - U >

The averages of the usual conservation laws ( I I I . 13) for NLS are obtained by expanding this expression (IV .11) in the local coordinates

5 = E 1 near E = <*> , denoted as <3 > + < £ > , J il 2 • There 2N are 2N variables { E^(X,T)} ^ , so we presumably choose any 2N of

these, arriving at the system of 2N , quasi linear, first-order

P.d.e's. Next, we investigate the 2N modulation equations. 47

IV .2 An Invariant Representation of the Averaged Generating Functions

For KdV , sinh-Gordon and s-G , Ercolani, Flaschka, Forest and

McLaughlin [1, 3, 5] have found a remarkable connection between averaged generating functions and certain Abelian differentials on these Riemann surfaces, from which they completely analyzed the modulation behaviors. This connection also exists for NLS , as we now describe.

We now define two unique meromorphic differentials on the Riemann surface ft of R(E) :

(k) ik) ‘ dE Definition: a = f I D. EJ(-mrc , k = 1,2 , (IV.12a) j -0 J RlEI where

(IV.12b)

( 2) ( 2) ( 2) 12 D = 4 , D = -2 YE , D = [- - ( 7E ) +2 TEE], N+l * N f k ’ N -l L 2 V f k A l j 1 1 1>J

(k) (ii) / n = 0 , k = 1,2 , 1 < j < N -l . (IV.12c) a . — — J

(k) (k) (k) We note that D.. . , D.. , D., . , k = 1,2 , are taken to be N +l N N - l (±) (±) (±) DI1L1 , D.. and D.. . , (IV .7, 8), for the averaged generating N +l N N - l functions. 48

(k) I t turns out that a , k = 1,2 , are fundamentally connected

to the averaged generating functions, the wave numbers k and

frequencies w of q.. , the averaged conservation laws and N modulations of q^ , and the linearized instability of q^ .

Lemma IV .3 (Invariant Representation of Averaged Generating Functions)

fi(1) = dE . (IV.13a)

( 2 ) a 2 <4Ef + i(qh + rg)>dE . (IV.13b)

Some Remarks: (1) (IV .13) shows that the averaged generating

functions- . . actually4. „ A define a , and ^ a ^ are unique. We use

linear algebra to show (IV .13) in Appendix G . (2) From (IV .7, 8), we

find

(k) + (a) a has a pole of order k+1 at * , « on ft , and

no other poles;

(k) -1 (b) The expansion of a in the local coordinates E = 5 ± near ® are:

a ^ ~ ± [- — d? + holomorphic part] (IV. 14a)

_ 5 1 ± , E = — near « . 5 (2) -4 a ~ ± [— ds; + holomorphic part] (IV. 14b) w %

In Appendix H, we use Riemann bilinear identity for the Abelian

differentials of first and second kind [68] to show 49

Lemma IV .4 (Physical Contacts of Averaged Generating Functions)

(i) /. fi(1) = -k. (IV.15a) bo J , 1 <_ j £ N-l .

( 2 ) (ii) n = -a). (IV.15b) bo 0

Remark: Previato [54] showed

K JE n(1) ~ E - i ~ + 0(E_1) , (IV.15c)

as E + oo .

(2) 2 -1 , /_ a ~ 2E + + 0(E ) , (IV.15d) N

IV .3 An Invariant Representation of the Modulation Equations;

Consequences of the Invariant Representation

We now consider

(D <2) « = nT - . (IV .16)

By straightforward calculation, we find

Lemma IV .5 (Basic Properties of n)

( 1 ) ( 2 )

r) rN^ , (1) (2), dE 1 2!) kT kX

(IV.17a) 50

(ii) Near E = — = » , the local representation of ft is:

0 0

ft ~ + { I [< Jf.>T - < X .>Y]'C**’ 2}d5 as E = — near »+ . p J • J a £ (IV.17b)

2 (iii) Near E - E = £ = 0 , the local representation of ft is: IV

+ holomorphic part] (IV.17c)

near 0 where

( IV. 17d)

We now use the Riemann-Roch theorem and Riemann bilinear identity to derive the main result with proof given in Appendix K, a method due to Flaschka, Forest and McLaughlin [1,3].

Theorem IV .6 (An Invariant Representation of the Modulation Equations)

Assume the fir s t 2N averaged conservation laws are satisfied by 2N {E|<(X,T)}1 51

<3 > + < Z > = 0 , j = 2,...,2N+1 . (IV. 18a) j T j X

Then all higher conservation laws are satisfied, and the modulation equations (IV.18a) take the equivalent form, via differentials on ,

Q = - a[2) = 0 . (IV.18b) I A

Theorem IV .7 (Consequences of the Invariant Representation (IV .18))

(i) (Riemann Invariants): The invariant form (IV.18) in equivalent to the Riemann invariants,

(k) E - S E = 0 , 1 j< k £ 2N , (IV.19a) T KX where

N+i . N+l s - [ I Dj 'e?]/[ J dJ1'^] . (IV.19b) j=0 J K j=0 J K

( i i ) (Modulational In s ta b ility ): The modulation equations are e llip tic , and the modulational in stability is predicted for all quasi periodic NLS wavetrains.

( i i i ) (Conservation of Waves):

( k ) - (to.) = 0 , 1 < j < N-l . (IV.20) J T j X ~ - 52

Remarks on the Proof: (i) The equivalent Riemann invariance (IV .19) is a immediate result of (IV. 17). ( i i ) Since , J J 0 < j < N+l , are real (IV .9), while E $ R , the characteristic (k) speeds S , 1 <_ k <_ 2N , are nonreal; consequently, the modulations are unstable, ( i i i ) The conservation of waves (IV .20) is due to (IV .15).

Remarks: for NLS wavetrains with periodic boundary conditions in x of fixed period L , there are N commensurability constraints on the parameters E ,...,E : 1 2N

L n ‘ — = , j = 1,2 N, n. € Z , (IV .21) a k, j J which reduce the system to N degrees of freedom. Clearly, k. must J a ll be identically constant or these commensurability conditions are broken. Consequently,

(k ) = 0 , j = 1,...,N . (IV.21a) J T

From conservation of waves, (IV .20),

(w.) — 0 , j - 1,...,N , (IV.21b) J * 53

i. e . , no spatial modulation in the presence of periodic boundary conditions for fixed period. Moreover, in the absence of external modulations, there are no terms to balance temporal modulations of

{E,} ; from (IV.19), k

(k) (Ek )T = - S (Ek)x = 0 , l

consequently, no modulations, temporal or spatial, are possible for x-periodic wavetrains of fixed period in the absence of external perturbations. CHAPTER V

LINEARIZED INSTABILITIES OF PERIODIC

N-PHASE NLS WAVETRAINS

Ercolani, Forest and McLaughlin [6] have completely analyzed the linearization about an N-phase s-G wavetrain by certain geometric methods; in particular, they showed exponential instability occured quite often. Independently, Tracy [53] used the inverse spectral theory to study these modulational in stabilities for the periodic NLS solutions in the neighborhood of the NLS plane wave. We now follow

[6] to investigate the N-phase NLS wavetrains. We will (1) connect the classical argument of linearization with Floquet theory for the

NLS plane waves; (2) show that the geometry of the NLS y-cycles characterizes the linearized in stability; (3) present an analytic argument to verify the geometric predictions.

V.l The Linearized Instabilities of the NLS Plane Waves

We now consider the NLS plane wave q , o

ia2t qQ = ae , a € R . (V.la)

We seek a neighborhood solution, 55

ia2t q = % + ql where qi = be » lbl << 1 (V.lb)

The linearized NLS about an arbitrary solution q is 0

+ qx + 4h 0|2ql + 2q2 . qx = 0 . (V.2a) t xx

For Qq.Qj in (V.l), we find

2 2 * {i a^ + 3 + 2a )b = -2a b , t xx (V. 2b) 2 * 2 (i a ^ + 3 + 2a )b = -2a b , t xx which decoupled into a simple equation of b ,

(3 + 3 + 4a )b = 0 (V. 2c) t t xxxx

Now, le t b(x,t) = ea t»e1*CX«b , and we find the linearized growth rate a :

2 2 2 2 a = -k (k - 4a ) (V.3)

2 2 k = 2a : most unstable. ★ 2 2 kQ = 4a : instability cut off.

Fig. 5 56

2 2 (V.3) shows long waves (0 < k < 4a ) are unstable.

We now impose the boundary condition with fixed spatial period L.

2 4n2 2 Then only a discrete set of unstable modes k = ----- — is allowed: n . 2

2 2 / ■ x 2ia t . ,. ^ , . . 2 2 4n ... (ll qQ = ae is linear stable for 4a < k = —- . (V.4)

2 2 ja ^ ( i i ) q = ae is unstable and there are 2n Fourier

2 4n2 2 exponentially unstable modes, k^ = — , provided

(v.5) a a

This classical argument shows the number of unstable modes is a function of the spatial period L .

On the other hand, we have shown, in Sec. I I . 4, for the NLS . 2 plane wave q = aeia with spatial period L , the double points z , {Ek € C , A(E^) = ±2 , A'(Ek) = 0} > are completely determined by L . I t turns out that the number of Fourier unstable modes and the number of nonreal double points are equal; moreover, each nonreal double point labels a unique unstable mode, as indicated in Figure 3 and (V.5). This is, essentially, the result of Tracy [16]. 57

We generalize the above in stability results and the connection with complex double points to arbitrary NLS periodic N-phase

solutions next.

V.2 y-Coordinates; A geometric Realization of Linearized Instability

We recall from Chapter II that for an arbitrary NLS solution q : (1) for generic q , there are no double points, Ed(q) ; (2) when q is degenerate, e.g ., an N-phase wavetrain q , the N (open and closed) degrees of freedom in q are in 1 : 1 N correspondence with the c ritic a l points of a ; moreover, when a closed degree of freedom is excited, the corresponding double point Ed splits into a pair of simple spectra, which induces an open degree of freedom. We also show in Chapter I I I that all open degrees of freedom, 2N N-l { E,}, in q1( are completely determined by {y .}. . In [80], k 1 N j 1 Ercolani, Forest and McLaughlin have shown that complex double points correspond to non-torus components in the isospectral set. We w ill not focus on this point of view here. So we assume the remaining y-variables are tied to the double points • Consequently, when we open certain closed degrees of freedom for q and investigate the N s tab ility problem, we may study the y-geometry instead. The homology between a-cycles and y-variables, (I11.42), allows us to use the simple closed curve {a.} . J We now open a simple double point E^ of £d(q..) to order e , d N Q where we have assumed the associated y-variable is tied at E. . Two 0 0 a d cases are considered: E . € R and E . f R • 58

0 0 Case 1: E, € R , with the corresponding u = E , as in Figure 6a. T ~ d d Ed splits into

(V.6) < ■ E“ + 0(e) ■ Ed • and the corresponding u -cycle is homologous to a a-cycle, as given in Figure 6b;

r ° - 0 Ed=lJ

Fig. 6a Fig. 6b

*: indicates E^ , E^ d d

E^ = E^ + 0(e) . d d

This picture indicates

e 0 |u ■ p | = 0(e) for both x and t flows, (V.7) which suggests linearized sta b ilitie s of both x and t flows. 59

0 0 Case 2: E . g R . Thesymmetry of the NLS spectrumshows E . is ------d d 0 0 also a double point, and thecorresponding y-variables arey^ = E^

0 0* 0 and y^ = , as given in Figure 7a. E^ splits into

E* = E° + 0(e) , Ee = E° + 0(e) . (V. 8a) I d 2 d

★ * * E^ splits into E^ and E^ . The corresponding y^ , y^ have homologous a1>a2 cycl es as given in Figure 7b.

o * ° - F ° V2=E6

Fig. 7a Fig. 7b

This picture, Figure 7, indicates

Iyk " ^1 = ^(1) » 1 i. i. 2 > (V.8b) which suggests linearized in stab ility. 60

Actually, as we w ill show in Section 3, such a perturbation to the degenerate NLS potential q„ yields modulational instability in the N t-flow . The geometric evidence is achieved by taking the linear combination of cycles corresponding to the x-flow and t-flow, as given in Figure 8a,b.

Fig. 8a: t-flow Fig. 8b: x-flow

V.3 The Analytic Argument of Linearized Instability

We now verify the geometric result that the degenerate NLS

N-phase wavetrains are modulationally unstable. Consider a NLS

N-phase wavetrain q (x ,t) , N < °° , with 2N simple spectrum 0 2N + 0 N-l {E } , N-l p-variables {p. and the hyperelliptic curve k l 2 0 (E) = n (E-E^) • °ur object is to compute the linearized growth rate corresponding to opening a closed degree of freedom (a double 61

point E ) to order e , and calculate the effect on the associated d dynamical y variable which, for e = 0 , is assumed locked at the

double point.

0 Case 1: E, g R : y,, = E, and E, breaks into E„.. . , j = 1,2, ------d N d d 2N+j

(1) * E2N+j “ Ed + eE2N+j * j = 1,2 ’ E2N+2 = E2N+1 * (V.9a)

with

- f 4. ( 1 ' “n - Ed + £“n

0 0 * Case 2: Ej j£ R ; ytl = E_, , y,,., h E_, . They break into ------d N d N+l d

E2N+j ' Ed + eE2N+j ’ j * 1,2 ’ and E2N+j ’ •* ' 1,2 ’ lV ‘ 9b) with

- F ♦ ( U - P * + 111 “ N - Ed + EI,N ■ 11 N+l ■ Ed + e|JN+l •

Now, the perturbed q , 6 = 1,2 , has 2N + 26 simple N+o spectrum { E > , N-l+5 open y-variables {y } and the associated 2N+26 curve = n (E - E, ) , where N+6 k 62

0 ( 1) Ek = Ek + eEk , 1 < k < 2N , (V. 9c) and

0 ( 1) y k = y k + e • yk , 1 < k < N-l .

We not insert (V.9) into the y-dynamical system (111.23), expand in e , and retain terms to order e :

2N+26 N—1+6 (° )

( 1 Ek' 2 1 " j) , 6 = 1 or 2 V h s J,P 1 U=0

n (p - p .) i*? p J

2N+26 N-l+5 ( ) ( I E^-2 J y“) w ; ,p

n (p - p .) P J J*P (V.10) for yN * PN+1

2N N-l ( ) ( I E°-2 I P°) (-20R („°) -1 ■ ^ ------e. ( I I I . 23), for P N-l n (y - y .) 1 < p < N-l . j * p p J 63

e ' term: we need evaluate R' (y )| and N+6 p e=0

2N+26 N-l+6 (*?)

< \ V 2 I ^ 1______J*P The results are N—1+6 II (y - y .) p J = 0

2N+26 y 1 - E ' P k ] (V .lla ) (1> W i V ’ p 1,1 1 y - E . k=l P k 6 = 1 or 2 .

2N+26 N-l+6 (J) ( I E - 2 I y ) ( i i ) [ ----- ?:------JfP------j. = ------1------L N-l+6 J N-l+6 n ( y - y . ) II (y -y .) P J P J J*P J*P

2N+26 N -l+6 . . ( , ) 2N+26 N-l+6 ( ° ) [( I -2 I »S ) -( I V 2 1 "d5 (V'Ub) 1 j*p 1 j*p

N-l+6 va ) -ua)

I - J ~ ] • j*P UP “j

We insert these quantities into e ' term with e = 0 , we find: Lemma V.l (Linearized y-Dynamical System)

111 2n n “- 1 » 0 m (u'1’) - (-21) ' / [(I E° - 2 I „ V

; (m° - u°) 1 j,k . . k j j*k

(1) (1) (1) (1) (1) r l y ^k " E1 N"x yk " yj „ yN-l+6 , l 2 0 r0 " }. 0 0 ^ 0 0 ] * - 1 , k - Et J .k „ k - u . I , k - W l

(V-12a)

2N (1) N_1 (1) (1) ^ + ( I E - 2 I y . - 2 I y , ) ] , 1 < k < N-l . , i r, 3 L N-l+6 J - - 1 J*k 6

2N 0 0 ( I E - 2 I y + 2y ) (1) 0 1 1 w i o S-L>X ■ -- n-"i ■■------1

* S-l« - "? J=1 (V.12b) ( 1) yN-l+6 ’ 0 0 * where y = E (and y = E ) , 6 = 1 or 2 . N d N+l d

Remark: y^,X] are decoupled with other y^1^ terms. Moreover, ------N-l+6 J

(III.2 2 b ,c ) and (V.12b) show y |X\ x ) and y.^j(x) move with complex N N+l conjugate pattern under complex conjugate in itia l conditions, i.e ., ( 1) * ( 1) (yN (x)) = uN+1(x), an analytic evidence of Figure 8.b. On the

other hand, ykl^ ( t ) and y ,* |( t ) do not move in such a pattern, as N N+l 65

evaluated from the t Floquet exponent of (V.12b), which indicates the

, v.,W , *t-flows xi • in r- Figure 8.a. N N+l

We now investigate the behavior(s) which is related to

the Banjamine-Feir in stab ility (see [6 ]). By applying classical

Floquet theory, we evaluate x ,t Floquet means instead of Floquet exponents,

"N-l+6 1 ("2l)RN(l,N-l+S1''N-l q ^ ’ (V. 13a)

I (v i « - v

w (t): o - - - o >

N -l „ (V' 13b)

1 “j - 2<------1 - J ------>] “- 1, o o. J

where <• > is the multiphase averaging usued in Chapter IV. The linearized flow is stable i f f the corresponding Floquet mean is purely imaginary; the linearized growth rate is given by the real part of the corresponding Floquet mean.

Ercolani, Forest and McLaughlin [6] have shown the Floquet means ■f of s-G linearized u-variables can be represented as integrals of 66

certain unique Abelian differentials. Based on their method, we also develop a technique to evaluate the Floquet means with the same results:

2 Theorem V.2 (The Floquet Means as Integrals of ft )

The x-flow Floquet mean, (V.13a) = if

The t-flow Floquet mean, (V. 13b) = if

We leave the derivation of (V.14) in Appendix J.

The integrals (V.14) are nontrivial as evaluated in Appendix K, and the results are:

Theorem V.3 (Modulational In stability of N-Phase NLS Wavetrain)

(i) (Linearized x-Flows are Stable)

Floquet exponents of (x) are purely imaginary and are equal to ik..^ ; uf,1 ? , x-flows are stable. N+o N -l+o

(ii) Linearized t-Flows are Stable iff E,€ d R .

When E R , the linearized growth rate is real and d non-zero, given by

(V.15) 67

* where y is a path from EJ to EJ on the fir s t sheet of the d d d ( 2 ) Riemann surface ft such that ft does not vanish. N

Remark: as mentioned in Chapter I I , there are at most fin ite ly many

nonreal double points. These results can be summarized in

Theorem V.4

Given q^ , an N-phase periodic NLSsolution, and its

associated double periodic spectrum • Let p = the number of * d ^ complex pairs of double points, z (q„) (1 {C\R} = {E',E‘ ,...,E^,E^ }. N d d d d Then q has 2p linearly unstable modes, with growthrates given by N

Imt / (£ ,—R ) ] = ’ Ed 6 {Ed,Ed Ed’Ed } * (V,16) d N a ★ where y, is a path from EJ to EJ as in Theorem V.3. Moreover, d d d a k i f Re(E.) = R e(Ej , 1 < £,k t-growth rate of E^ i f f Im(E^) > Im(E^) . d d d d CHAPTER VI

CONCLUDING SUMMARY; FUTURE RESEARCH

V I.1 Concluding Summary

The Riemann-theta function representation ( I I I . 39) of the inverse

spectral solution qtl which yields 2N simple spectra {E, } shows N k q„ has N degrees of freedom and is parameterized by N real phases N ------{e .} with 2ir-period in each phase e. . The physical wave number k. J J J and frequency w. of e. are the integrals of unique Abelian J J •I' differentials over b.-cycles, i.e., (IV.15). J

The u-representation of Un q ) , (III.24), along with the N X t Abel-Jacobi inverse map shows qN is also parameterized by the N-l

D irichlet eigenvalues { y .} and the plane wave phase e . J N While those phases {0^.} precisely describe the behavior of q^ ,

the {y .} variables are fundamental to the perturbations of ql( . j N

The modulation of the modulating N-phase NLS wavetrain q.,(e;E) N is prescribed by 2N modulation equations, the fir s t 2N averaged

conservation laws. The homology, a. ~ y. , simplifies the J J .

averagings. Due to the connection, (IV .13), between s? v and the

68 averaged generating functions, the modulation equations are then

analyzed by n and represented in a invariant form (IV .18), ( 1) (?.) 2 fl = = 0 . Expanding n = 0 at E - E = 5 ~ 0 , the I A K Riemann invariants (IV .19) are derived. Evaluating / . a = 0 , the b . J conservations of waves, (IV .20), are obtained. For x-periodic , N there are no spatial or temporal modulations without external

perturbations.

To analyze the linearized in stab ility of the exact N-phase wave train qti , double points EJ are opened up (to order e) and N d the linearized y-dynamical equations are investigated. The Floquet ■f means of the linearized y-equations are shown to be the integrals of

< 2 > n , (V.14). From which, the unstable modes are identified and associated to the nonreal double points E, , and all linearized d growth rates are explicitly calculated, i.e., (V.16).

V I.2 Future Research

Numerical studies by Overman, McLaughlin and Bishop [64] have conclusively indicated that the damped, driven sine-Gordon equation,

q - q + sin q = et-aq^ + r cos(

Grassberger-Procaccio dimension of the attractor. The spatial coherence is quantified via a precise measurement of the s-G periodic mode content in the spatial profile at each instant in time. That is, the fu ll Pde (V I.1) is integrated numerically and, at each time step, the resulting q (x ,t ) is placed into the Takhatajian-Faddeev n eigenvalue problem, from which M E ;q (x,t )) is computed numerically, n and then the periodic spectrum is calculated. Even in parameter regimes of e , a , r where the t-flow is chaotic, there are only a low, fin ite number (2-8) of s-G modes excited.

The natural questions is: how can an integrable equation, under small perturbations, become chaotic? According to modern dynamical systems theory, existence of homoclinic orbits in the unperturbed problem is a setup for chaos under small perturbations. Following the analysis of Ercolani, Forest and McLaughlin on sine-Gordon [80], we want to use Backlund transformations to explicitly find homoclinic orbits in the periodic NLS phase space. These orbits are realized as the global p-paths associated to complex double points. We w ill then investigate the perturbations of these orbits, under damping and driving, via the Melnikov technique [81]. This work will require extensive numerical calculations. APPENDIX A

PROOF OF ( I I I . 22)

* Lemma: Supposef. € R and g. = -h . , then J J J

(i) gi i = hi i * (A,1) J J O < j <_ N . (ii) f. , € R , (A.2) J -l

Recall (III.1 7 a ), g = -2ig + 2qf , h = 21h - 2rf , jx j-l j jx j-l j ^ ^ ^ r = q . This implies g. + h. = -2 i(g . + h. ,) + 2 q (f. - f .) , jx jx 3j-l j-l J J i.e., (A.l).

To prove (A.2), we recall ( III.18),

2N 2N f 2 - gh = n (E - E ) = IK E* . (A.3) 1 K 0 £ ★ Since E = E , consequently, K € R , 0 _< £ £ N . N . £ N-l N-l Expanding f - gh with f = I f.E J , g = I g.EJ , h = J h.EJ , 0 J o J o J (III.16), we find

K0.. . = 2f f . + I f f + I (g h + g h ) € R . (A.4) 2N-J N N-j L k £ L £ k k £ N-j < £,k £ N-l N-j < £,k £ N-l k+£ = 2N-j £+k = 2N-j

71 72

★ Suppose f f . e R and g = -h , N-j+1 < z < N-l , then, N N-j+1 z Z * ~ ~ since € R and g^hR + gRh^ = -2Re{g^gR) , N-j+1 <_ £,k :: N-l, we find, from (A.3,4) , f . € R , i.e., (A.2) . N-j

We now combine (III.1 9 a ) and (A.1,2), and use Mathematical

Induction, so (III.2 2 a ,b ) is completely verified.

We now prove (III.2 2 c ). Recall ( I I 1 . 16, 19a, 21),

N-l N-l g. g =(-iq ) n (E-u ) = g ( I Ej ) where g = -iq . (A.5) 1 3 0 9N-1

Then, for real E ,

* * N_1 * N-l h. -g (x,t;E) = -Iqn (E-„.) = h ( I — i - EJ) . (A.6) 1 J ,l' 1 0 N-l ★ Here, we use the fact g. = -h. . Consequently, for real E , J J

* N-i N-l * -g (x,t;E ) = h(E) = I h.EJ = -ir n (E-u.) . (A.7) 0 J 1 J

Now, for complex E ,

* N_1 * * * * N_1 * g(E ) = (-iq ) n (E -u .) , g (E ) = iq n (E-u.) , (A.8) l .e .,

N"1 -g (E ) = - i r n (E-u .) = h(E) . (A.9) 1 J 73

Here we use the tact that h is continuous in E . This completes the proof of (III.22). APPENDIX B

PROOF OF (111.23,24)

Proof of (III.2 3 a ): recall (III.1 7 a , 21),

g = 2qf + (-2iE)g , (B.l) X N-l g = -iq n (E-u ) . (B.2) 1 K

Inserting (B.2) into (B.l) yields

N-l N-l N-l (-iq ) n (E-u. ) + (-iq) J • n (E-y.)] * 1 K 1 * j*k J (B.3) = 2qf + (-2iE)q .

Let E = u in (B.3), (III.2 3 a ) is obtained. P

Proof of (III.2 4 a ): from (111.16,21), we find

9N-2 = 1q * (B‘4)

From (III.19a),

74 eeea^iiAx

N-2 2

(B.4) and (B.5) yield (III.24a).

Proof of (III.2 3 b ): recall (III.1 7 b ),

g = 2(iq + 2Eq)f + 2i(qr - 2E2)g . (B.6)

Inserting (B.2) into (B.6) yeilds

N-l N-l N-l (-iq ) n (E-u.) + (-iq) J (-u,) + n (E-u.) 1 1 3 1 0*k J (B.7)

= 2(1q + 2Eq)f + 2i(qr - 2E )g . X

Let E = u in (B.7), we find P

2 <1qx + V Up ) t = TN-l T - r ^ ------iq

Inserting (III.24a) into (B.8) yields (III.23b).

Proof of (III.24b): recall (III.17b, 19a, 21), gN = -iq ,

9N-3 “ ( " 1 q l i . Vi > 1>J

qt = i(9N-lJt = 1I2iqxVl + 4qfN-2 + 21qrVl - 4l9N-3] •

HOLOMORPHIC AND ANTIHOLOMORPHIC INVOLUTIONS: PROOF OF ( I I I . 29, 30)

We define the antiholomorphic involution a on the Riemann

2 2N surface (E, R (E)) , R (E) = n (E - E ) , 1 k ★ ★ cx(E,R) = (E ,R ) . (C .l)

2 * * Clearly a = identity and R (E) = ±R(E ) . Either taking + or -

signs gives the same answer, we now take + sign. Next, we define

"pull back" of the differentials { dl).} . Recall ( I I I . 26), J

N-l N-k-1

dUj * [ TuTdE ■ (c’2' then the pull back is

N-l , * N-k-1 * (E ) * a (dU.) = Y C., - —T------dE . (C.3) J 1 Jk ME*)

Consequently,

N -l N-k-1

,dv> ■ [ c# W * • (c-4)

77 78

Let y be a one cycle on the Riemann surface. Then

■ W #*(dV)* ■ (c-5’ which can be seen from the Riemann sum of the integral. Since dU. is * * ^ holomorphic, so is (a (dU.)) , which then can be expanded on a J holomorphic basis. We recall "a-b cycles" on this Riemann surface.

From the cut structure of E-plane and a-b cylces construction, as in

Figure 4, we find

N -l a(a.) = -a. , cr(b ) = b + Y a.. (C.6) J J k k . j j * k Recall ( I I I . 26, 27), we find

* N-l N-k-1

(U.1 dV 1 dUj “ ij */.. 1 I 1 Cjk W dE * (C'7al and

(/ dU .)*=/, . (a*(dU .))* = / (a*(dU .))* . (C.7b) j ja(a.) j J-a j

From (C.3, 7) we find

cj k = -cjk • 1'- e- cj k 6 1R • ,c -81 and ★ ★ dU. = -{a (dU.)) . (C.9) J J 79

This verifies (III.29), i.e., (C.8).

Recall ( I I I . 28), B.. = f. dU. , then ij Jb. j

Vl u/ = U > * (dV ) % ' N-l <'*tdU/ 1 b. + I a i • £ (C.10) N-l = / dU - f dU = -B - J 6 bi j v d ij Wi l . a«

Consequently,

* N-i (B ..) = - B.. - I 6 . , (C.ll) ij ij . £*1 i.e ., B.. e iR and Re(B..) = - — for i * j , which is JJ iJ 2 ( I I I . 30). APPENDIX D

PROOF OF ( I I I . 33)

Proof of (III. 33a) : recall ( I I I . 26, 32, 23a). We now calculate

U.) , J x

N-l-v N-l N-l U .) = I C. f I (-21) -----7 r (D.la) J x J j / j n (uk-iip) p*k

N-l-v N-l N-l = C-2t) I C ( I (D.lb) N-l ) • 1 J 1 * p*k

The Lagrange Interpolation formula is

N-l-v N-l (D.2a) N-l Vl-v,N-2 • 1 ± N'1-V i N-2 * k=l n„ W p*k 1 if v = 1 , (D.2b) 0 otherwise.

From (D.l, 2), we have verified (III.33a), i.e.,

U .) = -2iC . (D.3) J x jl 80 81

Proof of (III.33b ): recall (III.23b). We now calculate U_

N-l-v N-l N-l y (D.4a) “ A ■ \ cjvt \

N-l N-t 2N N-l ■ (-2i> I CJV[ I (I h'2 1 V1 (D.4b) 1 J l , , l t*k

p*kV v V

N-l-v 2N N-l N-l E )•[ I C • ( (D.4c) <-21M I I N-l )] 1 1 J 1 V yk V p*k K

„ , N-l N-l-v „ N-v v,. • I v.- uk N-l N-l k + ( 4 i ) . i c . r y ------i J jVL J N-l J n (yk -y ) p*k

N-l N-l-v 2N N-l N-l Hk ■ -M( I E£)Cdl ♦ 41 . I C.v I J ] (D.4d) N-l

" A - y p*k K

N-l-v N-l N-l 4i y c. r y — ------1 , jv L t N-l J 1 JV 1 n (yk-y ) p*k 82

2N N-l N-l -2i( I E )C +4i(y y) JC 6 ^ ZJ j l * y L jv N-l-v,N-2

N-l N-l - 41' V ? in ) V v V p*k

N-v N-l N-l -) (D.4f) 41 * CJV (\ m

p*kV v V K

We notice that the last term in RHS of (D.4f) is for v 2 , so

N-v _< N-2 , and the Lagrange Interpolation formula applies. We find

2N N-l U ) = -2i( I E )C + 4i( I y.)C J t j « Jl x J Jl (D.5a)

N-l y^"1 N-l

- 4i V \ m - ) - 4i l V V v ,N -2

V - k V p*k

2N = -21( y E ) C. - 4i C . (D.5b) x H Jl J2

This verifies (III.3 3 b ), i.e ., (D.5b). APPENDIX E

PROOF OF THEOREM I I I . 9

First, we evaluate 3 • Recall (III.44b),

N-i y 6j - Z /„*

Let y be the path from y to y as given in Figure 9.1. k k

Y : from y^ to y^ .

Fig. 9.1

We use the integral technique in Appendix C and find

83 84

(/ dU.)* = / (a*(dU.))* = -/ dU. = / dU. , (E.2) Y J M y ) J -Y J Y J i .e .,

/ dU € R , (E.3) Y J which implies, from (E.l), 6. 6R . J

Now, we evaluate d . Recall (III.3 9 f),

d. = f dU. , 1 < j < N-l (E.4) J - J “ “

Let y be the path from «. to « as given in Figure 9.2,

<—

: upper half plane. : lower half plane.

Y : from ~ to <*>

Fig. 9.2 then 85

N-l ; + du. - i i- du. ♦ j N_1 du. (E.5a)

N-l

= \ <- l )8k j+ ^ dUj - - 1 + / Tl dUj •

N-l j where y, = y - I (- ~)a is a real path. 1 ^ C n

(/ dU-) = / / v - du. = -/ dU. , (E.6) w yx y j o(yi ) j j i .e.,

J dU. € iR , (E.7) Y1 J

+ 00 1 from which (E.5b) implies Relf dU.) = — . 00

We now evaluate Re(t. - j g.) . Recall (III.39e, 44b), J t J

j = i 6jj - V U dUj'd\ + Y dUj ■ 1 1 J i “‘I • K-B> 1 K “ 1 00

N-l y = I J0* , 1 < j < N-l . (E.9) 1 “k 86

Then

;j - i s i y - V u t/P+ dV dUk 1 K <*> (E.10)

N-l Re N-l ^ N-l yk + I / . ,'d U .+ I ( " dU.-^-y/'dU.. 1 00 + J 1 1 Re Do u.° J 2 1 1 u. O J 1 Re "k 1 \

N-l First step: we evaluate I j ( /p dll.) dU in (E.10). For each 1 ak «+ J k k ,

[J U\ «U.)dUk]* - / ,fA ( /P+ ^)dUk]}* . (E.ll) k °° k °°

+ Let y■ be any fixed path from °° to p , then

(E-n l = -f0(ak)f° * [(/ Y dV dUk])*

N-l .N-l-v * . " / , >[(/ , i dU,) - I C E -2 — ] (E.12b) k p J 1 kV R(E )

(“*

= / [f (-dU.)J(-dUJ , (E.12d) -a y j k k P KUMreMr-'

i .e .,

( U \

N-l Re pk Second step: we now evaluate Y f dU. in (E.10). For each k, ------, + J 1 » + o let y be the path from <*> to Re y^ as given in Figure 9.3,

Rey

o from to Re y.

Fig. 9.3 88

o Re p, dU. (E.13a) £ - - a (Y -K -i )a£)=y 1 J 2 £

dU. , if 3£ , £ = j (E.13b) J J ' 1 ' / du. , otherwise . (E.13c) ^ Yj J

The path y is a real path, as in (E.6),

/ dU. € iR . (E.14)

(E.13, 14) implies

N-l Re I Re( I f + dU ) b - or 0 . (E.15)

1 00

O O N-l p N-l p Third step: we now evaluate Y / dU. - — Y / ^ dU. in ------1 Pe ° J i J 1 Rc uk 1 "k o * (E.10). Fixed k , consider the paths y^ and y^ from p^ to o o o Re p^ and Re p^ to p^ respectively as given in Figure 9.4, 89

„ ° o from to Re uk i from Re y, to y, Y2 : k Mk Fig. 9.4

1 A / dU.- - / r dU. = / dU. - j f . dU. (E.16) J no j 2 J O * J Y o j i Re y 12 J and

(J dU.) = J - dU. = / - dU. = / dU. (E.17) y2 j7 M y2) j J Yj j

(E.16, 17) implies

/ dU - (E.18a) y2 j 2 (/V2 dUj ' \ V

(E.18b) ' 2 C/Y2 dUJ ' (/Y2 ‘A

= i Imff dU.) , (E.18c) y2 0 90

i . e . ,

dU, - -7 / dU. € iR (E.19) L J 2 J

l .e .,

o o N-l y U, 1 < j < N-l . (E.20) I (/n o duj ■ i C dy 6 1R • 1 Re y, yk

We now use the fact B.. € iR and (E.10, 12e, 15, 20) and find J J

n /O 1 x 1 (E.21) ReUj - 2 6j ‘ S 2 0r ° APPENDIX F

PROOF OF (IV .7, 8, 9)

N-l By ovserving (IV.5), we need to evaluate < n (E-y.)> , 1 j N-l N-l N-l N-l N-l N- , , <( ][ y )2 . I (E-y )> , 1 J 1 J l>j 1 J 1 J 1 J 1 J N-l N-l . 1 J 1 J

Lemma F.l (A Useful Formula)

Let G be a gx g , g = N-l , matrix with G. . = y^ J . Then ij i

9 *1 j2 i g n (u.-u.) = det G = I sign(i i )y y ...y 9 (F.l) . ^ * J /. . « 1 a i d g KJ ( i ,...... i ) 3 3 1 9 where(^ . . . . . i ) is a premutation of (g -l,g -2 ,...,2,1,0) , and sign is positive "+" if even, otherwise, negative .

Lemma F.2 (First Fundamental Formula for the Averaging)

9 * = a m / ^ (y.-y.) --- , g = N-l (F.2) det M t 1 J 9 a 10 n R(y.) • 1 J J=1

91 92

Proof of Lemma F.2: from (IV.2b, 6),

(F.3)

_ f / X d e t P (F.4) " I* <0 det M ^ a

[/ (.) det G ] 1 (F.5) det M n R(y.) 1 J

[f (•) n ] 1 (F.6) i j g J det M 3 n/ t n R(y.) 1 J

Lemma F.3 (Second Fundamental Formula for Averaging)

** 9 Assume that F is a function of y = {y }7 , then k 1

(F.7) | fj‘j •

2N where £ = ^ and p(?) n (1 - E^) , and {F^.} are given in the V I proof. BBiSSMKS-'-