Soliton Solutions to Calogero-Moser Systems

Master of Science Thesis

Soliton solutions to Calogero-Moser systems

Axel Reis Philip

Supervisor: Professor Edwin Langmann

Mathematical Physics, Department of Theoretical Physics School of Engineering Sciences Royal Institute of Technology, SE-106 91 Stockholm, Sweden

Stockholm, Sweden 2019 Typeset in LATEX

Akademisk avhandling för avläggande av teknologie masterexamen inom ämnesområdet teoretisk fysik. Scientiﬁc thesis for the degree of Master of Engineering in the subject area of Theoretical physics.

TRITA-SCI-GRU 2019:383

c Axel R. Philip, November 2019 Printed in Sweden by Universitetsservice US AB, Stockholm August 2018 Abstract

We develop a general method of generating M-soliton solutions to the AN−1 classical N-body Calogero-Moser (CM) systems of rational, hyperbolic and trigonometric type, for arbitrary M < N. This method is based exclusively on properties shared between the Lax representations of these systems, and in particular on their relation to a certain functional equation. The formulae are obtained from the real and imaginary parts of Bäcklund transformations between systems of the same type. We also extend the method for treatment of the hydrodynamic limits (N → ∞) of these systems (not including the hyperbolic type), by reformulating our N-body formulae partly in terms of density and ﬂuid-velocity ﬁelds. This is achieved by exploiting a connection between the CM Bäcklund transformations, and soliton solutions to the Benjamin Ono equation, which can be shown to exist virtue of the above-mentioned properties of the Lax representations. Lastly, though it is not included in our general method, we discuss a formula for the generation of M-soliton solutions to the more general elliptic CM systems of classical AN−1 type.

These developments generalize particular results, recently obtained by Abanov et al. [1] [2], pertaining separately to the rational and trigonometric Calogero-Moser systems. We likewise expect that our results indicate a promising route toward further generalization, for instance to include the elliptic CM systems, and for adaption to other kinds of integrable systems with Lax representations, such as the classical relativistic CM systems.

We illustrate and examine the typical behaviour of our CM soliton solutions, in several instructive and representative solution plots. We also provide brief reviews to the wide array of subjects pertinent to the discussion, including the CM systems themselves, their Kazhdan- Kostant-Sternberg construction by Hamiltonian reduction, Lax representations, the Olshanetsky Perelomov projection method, Bäcklund transformations, the pole ansatze to the KdV, Burger’s and Benjamin Ono equations, and solitons within many-body and hydrodynamic systems.

i Abstrakt

Vi utvecklar en generell metod för alstrande av M-soliton lösningar till klassiska AN−1 Calogero- Moser (CM) system i N kroppar, av rationell, hyperbolisk och trigonometrisk typ, för godty- ckligt M < N. Metoden är baserad uteslutande på gemensamma egenskaper hos Lax representationer av systemen, och då särskilt deras anknytning till en viss funktionalekvation. Ifrågavarande solitonlösningar erhålls som lösningar till real- respektive imaginärdelen av Bäck- lundtransformer mellan system av samma typ. Vi utökar även metoden till att kunna behandla M-solitonlösningar i de hydrodynamiska gränserna (N → ∞) av systemen (exklusive det hyperboliska systemet), genom att delvis omfor- mulera våra tidigare N-kroppsresultat i termer av densitets- och ﬂödeshastighetsfält. Det senare uppnås genom ett samband mellan Bäcklundtransformerna och Benjamin-Ono-ekvationen, vilken kan visas existera till följd av ovan nämnda egenskaper hos Lax-representationer av systemen. Slutligen, även om dessa för närvarande inte ingår vår generalla metod, diskuterar vi M- solitonlösningar till de mer generella elliptiska CM-systemen av AN−1 typ.

Sammantaget utgör detta en generalisering av speciella resultat från Abanov m.ﬂ. [1] [2], som hänför sig till det rationella- respektive det trigonometriska CM systemet. Vi håller det för troligt att vår metod och våra resultat anger möjlighet till ytterligare generalisering, till att omfatta exempelvis de elliptiska CM-systemen, och för anpassning till andra typer av integrabla system, såsom de klassiska relativistiska CM systemen.

Vi illustrerar det typiska beteendet hos solitonlösningarna i ett instruktivt antal världslinje- diagram, från vilka vi drar slutsatser av både kvalitativ och kvantitativ art. Vi ger även ett antal allmänna introduktioner till de många ämnen som berörs i det följande, däribland till Calogero-Moser systemen själva, deras Kazhdan-Kostant-Sternberg uppbyggnad genom sym- metrireduktion, Lax representationer, Olshanetsky & Perelomovs projektionsmetod, Bäcklund- transformer, pol-ansatser till KdV-, Burgers- och Benjamin-Ono ekvationerna, samt till solitoner inom ﬂerkropps- och hydrodynamiska system.

ii Preface

Acknowledgements

I am grateful to Professor Edwin Langmann for his interested supervision and helpful advice, as well as for introducing me to the subjects on which this thesis is based.

Outline

This thesis is divided into two major parts. The ﬁrst part, comprising chapter 2, provides the theoretical background necessary to the discussion in the second part. In this part we review and list important results pertaining to the Calogero-Moser systems, their Kazhdan-Kostant-Sternberg construction by Hamiltonian reduction, Lax representations, the Olshanetsky-Perelomov projection method, Bäcklund transformations, the pole ansatze to the KdV, Burger’s and Benjamin Ono equations, and solitons within many- body and hydrodynamic systems. The second part comprises chapter 3, and concerns soliton solutions to the Calogero-Moser systems. Herein we derive general equations for the generation of soliton solutions to a subset of many-body classical Calogero-Moser systems of AN−1 type, and to their hydrodynamic generalizations. In relation to this, we also illustrate the typical behaviour of these soliton solutions in a number of worldline diagrams. A few relevant derivations and proofs are not included in the main text, but are instead relegated to the appendices. In particular, the proofs on which our general derivation of soliton solutions is based, can be found in appendices B.1 and C.3. The structure of the thesis is explained further at the end of the introduction in chapter 1.

iii Glossary of technical terms and symbols

• I, II, III, IV: Labels the Calogero-Moser systems of, respectively, rational, hyperbolic, trigonometric, and elliptic class, see §2.2

• I+: Labels the rational Calogero-Moser system subject to a harmonic external potential, see §2.2

• Auxiliary system: In chapter 3 refers to the Calogero-Moser system that governs the soliton solutions to a principal Calogero-Moser system.

• Calogero-Moser systems: Refers, for the most part in this thesis, to the classical Calogero-Moser systems of AN−1 type, see §2.2. • Hamiltonian reduction: See §2.3.

• KKS construction: Method of constructing certain Caloger-Moser systems by way of Hamiltonian reduction, see §2.3.

• Lax pair: A pair of operators or matrices L and M that satisfy L˙ = [L, M] and often comprise a Lax representation, see §2.4

• Lax representation: An alternative formulation of a dynamical system, in terms of linear operators of matrices, see §2.4.

• OP projection method: Method of explicitly integrating certain Calogero-Moser systems, see §2.5.

• Principal system: In chapter 3 refers to the Calogero-Moser system in which solitons appear.

• Solitons: Nonlinear wave-phenomenon, see §1 and §2.7.

• Special functional equation: α(x)α0(y)−α(y)α0(x) = α(x+y) α0(y)−α0(x) , where α(x) is a special Lax function.

• Special Lax function: An odd function α(x) is a special Lax function of the CM system with interparticle potential V (x) = α(x)α(−x) if α() = c−1/ as → 0, 0 2 α (x) = −α (x)/c−1 + const., and if it satisﬁes the special functional equation.

iv Contents

Abstract ...... i Abstrakt (på svenska) ...... ii Preface ...... iii Glossary of technical terms and symbols ...... iv

1 Introduction 1

2 Calogero-Moser (CM) Systems 5 2.1 Chapter summary ...... 5 2.2 Introduction to systems of Calogero-Moser type ...... 8 2.3 The Kazhdan-Kostant-Sternberg (KKS) construction of CM systems . . . 11 2.4 Lax representations ...... 14 2.4.1 Lax representations of many-body Hamiltonian systems ...... 14 2.4.2 Lax representations of isolated CM systems ...... 15 2.4.3 Examples of Calogero-Moser Lax representations ...... 17 2.5 Olshanetsky Perelomov (OP) Projection method ...... 19 2.5.1 I: Rational CM system ...... 19 2.5.2 I+: Rational CM system with external harmonic potential . . . . 20 2.5.3 II: Hyperbolic CM system ...... 21 2.5.4 III: Trigonometric CM system ...... 21 2.6 Bäcklund transformations ...... 22 2.6.1 General discussion ...... 22 2.6.2 Bäcklund transformations of Calogero-Moser systems ...... 24 2.6.3 Examples of BTs for CM systems ...... 25 2.7 Hydrodynamics and Solitons ...... 27 2.7.1 Calogero-Moser solitons in hydrodynamic systems ...... 28

3 Soliton solutions to Calogero-Moser systems 30 3.1 Solitons from Bäcklund transformations of CM systems ...... 32 3.1.1 General procedure ...... 32 3.1.2 I+: Rational model on the line ...... 33 3.1.3 I+: Examples of soliton solutions ...... 35 3.1.4 II: Hyperbolic model ...... 41 3.1.5 III: Trigonometric model on the unit circle ...... 42 3.1.6 III: Examples of soliton solutions ...... 43 3.2 Solitons from semi-hydrodynamic Bäcklund transformations ...... 48 3.2.1 Bidirectional Benjamin Ono equation ...... 48 3.2.2 General procedure ...... 49 3.2.3 I: Rational model ...... 50

v CONTENTS CONTENTS

3.2.4 III: Trigonometric model ...... 51 3.3 Concluding remarks on soliton solutions to the elliptic CM systems . . . 52 3.3.1 Summary of the uniﬁed method ...... 52 3.3.2 Soliton solutions to elliptic CM model (IV) ...... 53 3.3.3 Hydrodynamic soliton solutions to the elliptic model? ...... 54 3.3.4 Elliptic projection method and KKS construction? ...... 54

A Appendix on Lax representations. 55 A.1 A formal construction of the Lax pair of an integrable system ...... 55 A.2 Derivation of the CM Lax representations from Calogero’s ansatz . . . . 56 A.3 Local analysis of the functional equation and derivation of the general CM potential ...... 57

B Appendix on Bäcklund transformations 59 B.1 Proving the form of the CM Bäcklund Transformations (CMBTs) . . . . 59 B.2 Derivation of the CMBTs from the Schrödinger equation ...... 62

C Appendix on hydrodynamics and solitons 64 C.1 Some facts about the Hilbert transform ...... 64 C.2 Proof of theorem 2.7.1 ...... 64 C.3 Proof of theorem 2.7.2 ...... 66

Bibliography 68

vi Chapter 1

Introduction

To the classical Calogero-Moser (CM) systems belong a selection of dynamical systems, comprised of N interacting bodies in 1 spatial dimension, with the added virtue of being integrable. This includes some particularly simple examples of interacting particles in one dimension, such as N particles on the line interacting repulsively via an inverse cubed −3 force F ∝ [qi − qj] , as described by the Hamiltonian

N N 1 X X0 1 2 H = p2 + W (q ) + g2 , (1.1) rCM 2 i i q − q i=1 i,j=1 i j where g > 0 and the 0 on the sum implies that j 6= i. A further example is the Calogero- Sutherland system of N particles on the circle, interacting through an inverse trigono- 3 metric force F ∝ cos(qi − qj)/ sin (qi − qj), as obtained from

N N 1 X X0 1 H = p2 + W (q ) + g2 . (1.2) tCM 2 i i sin2 (q − q ) i=1 i,j=1 i j The examples, in the absence of any external potential (whereby W (x) = 0), constitute Liouville integrable systems; and (1.1) is known to be Liouville integrable with a harmonic 1 2 2 external potential included, W (x) = 2 ω x . They are however not integrable for arbitrary W (x). The aim of this thesis is the derivation of soliton solutions to the Calogero-Moser systems. The term soliton here refers to a phenomenon, present in nonlinear media, of localized waves bearing permanent form and propagating at constant speed. Moreover, the form (and hence identity) of the solitons should remain the same even after soliton- soliton collision, though the colliding soliton waves would then generally acquire a phase shift. The solitons might thus to some extent be characterized as waves with particle-like behaviour. The ﬁrst analytically derived soliton waveform would in present day be recognized as the hyperbolic 1-soliton solution 1 1√ u(x, t) = v sech2 v(x − vt − x ) , (1.3) 2 2 0 1 to the Korteweg de Vries equation ut −12cuux +cuxxx = 0. This solution is meromorphic, with an inﬁnite line of poles, pn(t), that is parallel to the imaginary axis and centered

1complex analytic except for isolated points which are poles

1 A. R. Philip Soliton solutions to Calogero-Moser systems

evenly about the real line [25] iπ 1 p (t) = x − vt − √ n + , n ∈ Z. (1.4) n 0 v 2 Moreover, its evolution in time is determined by the motion of these poles (or rather the motion of the pole-line along the real axis). This pole-evolution in turn is determined by a 1-particle Hamiltonian system (that is periodic along the imaginary axis), as was established by H. Airault, H.P. McKean & J. Moser in [9] and G.V. Choodnovsky & D.V. Choodnovsky in [3]. We provide a more complete review of hydrodynamic solitons in §2.7.

Soliton solutions to (AN−1) Calogero-Moser systems have been derived previously for two special cases, the rational CM system with external harmonic potential on the line and the trigonometric CM system on the circle, respectively, by Abanov, Gromov & Kulkarni in [1] and Abanov, Bettelheim & Wiegmann in [2]. The former is given by (1.1) with 1 2 2 W (x) = 2 ω x and qi ∈ R, and the latter by (1.2) with W (x) = 0 and qi ∈ [0, 2π)/2πZ (angles from 0 to 2π). Waves within these discrete systems of particles in one dimension must, of course, be longitudinal and generated by some pattern of coordinated concentration and dilution among the particles in the system. Hence they are waves of particle-density. A soliton wave is in addition permanent, wherefore its associated particle motions must be highly interdependent at all times and belong to only a restricted part of the phase-space available to the system. Indeed, as will be shown in §3.1, the soliton solutions effectively reduce the number of degrees of freedom of the system, in the sense that, for every initial soliton-wave configuration, the continued evolution of the soliton solution can be encoded in an auxiliary many-body system of fewer degrees of freedom. This should be compared to the hydrodynamic soliton solutions mentioned above, and discussed at depth in §2.7, whose evolution is encoded in a finite many-body system of poles. In [1] the ’reduction’ is achieved, for the rational CM system with harmonic external potential in N real coordinates xi ∈ R, by the ’soliton equation’

N M X 1 g X 1 ωxi = g − Re , (1.5) xi − xj 2 xi − zn j6=i n

g PM −1 together with momenta x˙ i = − 2 n Im [xi − zn] . The M < N coordinates zn ∈ C themselves belong to an auxiliary CM system of the same type, but defined on the complex plane. Solutions to this soliton equation are also M-soliton solutions to the rational CM system, as will be shown in §3.1. The soliton equation (1.5) does not possess any known analytic solutions, but in [1] it is solved approximately for M = 1. In this case, the single ’z1’ particle traces an elliptical trajectory on the complex plane (more generally each zn traces a different ellipse, so the zn system is not inherently one-dimensional), and an initial soliton configuration is determined entirely by an initial value z1(0) and the N equations of (1.5). We are at present able to reproduce the worldline diagram given in [1] with some additional detail, see fig. 1.1. In the figure, the worldlines to the principal system of particles {xi}, are represented by the set of black lines at iy = 0 (the real line). The blue curve around the principal system represents the elliptical trajectory z1(t), on the complex plane, of the auxiliary 1-particle system. As will be explained in §3.1.3, this auxiliary system can be considered as a positively charged particle on the complex plane, orbiting a system

Chapter 1 2 Section 1.0 A. R. Philip Soliton solutions to Calogero-Moser systems

of negative charges constrained to the real line. The trajectory of z1 then induces the soliton wave among the negative charges through electrostatic attraction.

0.8

π 0.6 t/2 ω 0.4

0.2

0 1.5 1 1.5 0.5 1 0 0.5 -0.5 0 iy -0.5 -1 -1 x/R -1.5 -1.5 Figure 1.1: World lines for 1-soliton solution to the rational Calogero-Moser system with harmonic external potential, comprising 40 particles on the real line (in black), with g = ω = 1. The soliton wave can be interpreted as being electrostatically generated by the elliptic 2D trajectory of an oppositely charged particle (in blue) starting at the complex coordinate z(0) = 1 + i (indicated by red dot), see §3.1.

Additionally, in §3.1, we oﬀer a number of graphs pertaining to soliton solutions with M > 1 auxiliary particles and soliton waves (wherein soliton interaction comes into play), and also to soliton solutions of the trigonometric CM system constrained to the unit circle, as illustrated by ﬁg. 1.2.

3 2.5

2 2 t t 1.5 1

1 0 4 0.5 2 4 0 2 0 Im -2 0 -2 -4 -3 -2 -1 0 1 2 3 4 Re -4 -4 phase (a) ﬁg 1 (b) ﬁg 2

Figure 1.2: Worldlines for 3-soliton solution to the trigonometric Caloger-Moser system of 8 particles on the complex unit circle (in red), with g = ω = 1. (a) The soliton excitation can be seen as being electrostatically generated by the orbit trajectories of three particles of opposite charge (in blue), starting at complex coordinates exp{izn(0)}, where z(0) = [ 0.6(1 + i), 0.6(1 − i), 1.2(4 − i)]. In (b) is plotted the evolution of the real angles of the particles on the unit circle. See §3.1 for details.

In [1] and [2] are also derived soliton equations that apply to the hydrodynamic limits of the rational and trigonometric CM systems. The method used is essentially based on

Chapter 1 3 Section 1.0 A. R. Philip Soliton solutions to Calogero-Moser systems

a discovery by D.V. Choodnovsky [8] (1978), namely that the soliton solutions

N M X 1 X 1 u(x, t) = − , (1.6) x − x x − z i i n n

to the Benjamin Ono equation2

ut + 2cuux + cH[uxx] = 0, (1.7) are governed by a pair of separate and independent rational CM systems in respectively N particles {xi}, and M particles {zn}. An analogous statement for trigonometric type solutions was found by H. H. Chen, Y. C. Lee & N. R. Pereira in [10] (1979). The Benjamin Ono equation is independent of N and M, which allows the soliton equations to be re-derived from it after the limit N → ∞ has been taken, such that the principal CM system is fashioned into a liquid, described by ﬁelds of density and ﬂow-velocity.

∗ ∗ ∗

Present thesis is structured as follows:

• Chapter 2 introduces (§2.2) the Calogero-Moser systems and a few of their relevant properties, including (§2.3) their Kazhdan-Kostant-Sternberg construction by Hamiltonian reduction, (§2.4) their Lax representations, (§2.5) the projection method due to Olshanetsky & Perelomov, and their Bäcklund transformations (§2.6.2). The chapter concludes with solitons in integrable nonlinear hydrodynamic systems (§2.7), where we show that the CM systems are intimately related to the soliton solutions of the Burger’s Hopf and Benjamin Ono equations.

• Chapter 3 derives soliton solutions to the Calogero-Moser systems, from the background knowledge of ch. 2. Soliton solutions to many-body CM systems of rational, hyperbolic and trigonometric type (§3.1) are derived, and their behaviour examined in a number of representative worldline diagrams. In addition, soliton solutions to the hydrodynamic extensions of the systems are derived (§3.2). The chapter concludes with a discussion on soliton solutions to the elliptic type CM system (§3.3.2).

• The appendices contain an assortment of calculations and proofs. In particular, we show that the auxiliary function β(x) to the general CM Lax representation may be written β(x) = −α(x)α(−x)/c1 + const. in terms of the Lax function α(x) and its asymptotic coeﬃcient c−1. We also show that the Bäcklund transformations of the rational, hyperbolic and trigonometric CM systems exist virtue of the properties of the Lax functions belonging to their Lax representations (appendix B.1). For the same set of systems and again using only the properties of their Lax functions, we prove that the Bäcklund transformations govern the pole dynamics of certain solutions to the Benjamin Ono equation (appendix C.3), and that the dynamics take the form of the Bäcklund transformations mentioned above.

2 p.v. R ∞ f(z) H[f] denotes the Hilbert transform, H[ f ](x) = π −∞ x−z dz, where p.v. stands for principal value.

Chapter 1 4 Section 1.0 Chapter 2

Calogero-Moser (CM) Systems

2.1 Chapter summary

Below we summarize those principal conclusions and results, of present chapter, that are necessary to the discussion in chapter 3.

Systems of Calogero-Moser type (§2.2) The classical Calogero-Moser (CM) systems are described by a Hamiltonian of the form

H(q1, ...qN , p1, ...pN ), (2.1)

with qj and pj the positions and momenta of particles moving in one spatial dimension. In this work we restrict ourselves to the AN−1 type CM systems, with some external potential W (x), described by

N N 1 X X0 H = p2 + W (q ) + g2 V (q − q ), (2.2) CM 2 j j j k j jk where the 0 above the sum implies j 6= k, and the interparticle potential  1/x2, (rational), (I)   a2/ sinh2 ax, (hyperbolic), (II) V (x) = 2 2  a / sin ax, (trigonometric), (III)   ℘(x), (elliptic), (IV)

where ℘( x | ω1, ω2 ) denotes the elliptical Weierstrass function. Prominent special cases 2 include particles on the line interacting via a rational interparticle potential VI(x) = 1/x and subject to a harmonic external potential W (x) = 2ω2x2; and particles on the circle 2 interacting via a trigonometric interparticle potential VIII(x) = 1/ sin (x). We label the former system by I+

N N 0 1 X 2 2 2 2 X 1 2 + HI+ = pj + 2ω qj + g . (I ) 2 qj − qk j jk The systems of class I - IV with W (x) = 0, and the class I+ system, are Liouville integrable [14]. • The CM systems with W (x) = 0, we in the following will be referred to as the isolated CM systems as they are free from external inﬂuences.

5 A. R. Philip Soliton solutions to Calogero-Moser systems

Lax representations (§2.4.2) The isolated Calogero-Moser systems permit Lax representations in Lax pairs on the form   L = δ p + (1 − δ )α(q − q ),  nm nm n nm n m N (2.3) X  M = δ β(q − q ) + (1 − δ )α0(q − q ),  nm nm n l nm n m  l6=n

where the N × N Lax matrices L and M obey the Lax evolution equation L˙ = [L, M]. The functions α(x), β(x) in turn satisfy V (x) = α(x)α(−x), β(x) = −V (x)/c−1 + β0 and the functional equation α(x)α0(y) − α(y)α0(x) = α(x + y) β(x) − β(y) , (2.4)

where the constant β0 is arbitrary but c−1 is the asymptotic coeﬃcient of α(x) as x → 0. Note that, for any V (x), an odd Lax function may be chosen such that V (x) = −α2(x) 2 and β(x) = α (x)/c−1 + β0.

Special Lax representations of CM systems (§2.4.2) For systems of type I, II and III an odd Lax function may be chosen such that β(x) = 0 −α (x) + β0, and the functional equation then reduces to α(x)α0(y) − α(y)α0(x) = α(x + y) α0(y) − α0(x) , (2.5) which we refer to as the special functional equation. The Lax function we likewise term the special Lax function. Provided a special Lax function a CM Lax pair is given by   L = δ p + (1 − δ )α(q − q ),  nm nm n nm n m N (2.6) X  M = −δ α0(q − q ) + (1 − δ )α0(q − q ),  nm nm n l nm n m  l6=n which constitutes what we term the special Lax representation.

The Olshanetsky Perelomov projection method of solution (§2.5) The projection method allows the isolated systems of type I, II and III, and the I+ system, to be solved exactly for qn(t), from an initial phase proﬁle qn(0), pn(0). The qn(t) are obtained as the eigenvalues of a matrix that is directly integrable. For example, the solutions of the I system correspond directly to the eigenvalues of the N × N matrix with components t Xnm(t) = δnm qn(0) + pn(0)t + 1 − δnm ig . (2.7) qn(0) − qm(0) The same holds for the I+ system with respect to sin ωt ig sin ωt Xnm(t) = δnm qn(0) cos ωt + pn(0) + 1 − δnm . (2.8) ω ω qn(0) − qm(0) Analogous formulas for the type II and III systems can be found in §2.5.

Chapter 2 6 Section 2.1 A. R. Philip Soliton solutions to Calogero-Moser systems

Bäcklund transformations (§2.6.2)

Provided the canonical coordinates pj and qj are taken complex the subset of systems associated with the special functional equation permit Bäcklund transformations on the form  N M  X X  x˙ j = − i α(xj − xk) + i α(xj − zn) + iωxj,   k6=j n (2.9) N M  X X  z˙ = i α(z − z ) − i α(z − x ) + iωz ,  n n m n l n  m6=n l where N and M are independent. Note that ω = 0 for all but the class I (rational) CM system, where α(x) ∝ 1/x.

Pole ansatze of the Benjamin Ono equation (§2.7)

Provided a special Lax function α(x), with asymptotic coeﬃcient c−1, soliton solutions to the Benjamin Ono equation, ut+2cuux+c−1c H[uxx] = 0, are provided by the meromorphic functions

N M X X u(x, t) = i α(x − xj) − i α(x − zn) − iωx, (2.10) j n where, ω = 0 for all but the class I (rational) CM system. From the residue of the solutions inserted into the Benjamin Ono equation, as x → xi and x → zn respectively, it is found that the motion of these poles is governed by the Bäcklund transformations of (2.9).

Chapter 2 7 Section 2.1 A. R. Philip Soliton solutions to Calogero-Moser systems

2.2 Introduction to systems of Calogero-Moser type

A many-body integrable Hamiltonian system is said to be an AN−1 type Calogero-Moser (CM) system if it is expressible in the form

N N 1 X X0 H = p2 + W (q ) + g2 V (q − q ), (2.11) CM 2 j j j k j jk with interparticle potential V (x) in the classiﬁed set of meromorphic functions [12]  1/x2, (rational), (I)   a2/ sinh2 ax, (hyperbolic), (II) V (x) = 2 2  a / sin ax, (trigonometric), (III)   ℘(x), (elliptic), (IV) where ℘(x) is shorthand for the elliptic and doubly periodic Weierstrass function [18]

1 X 1 1 ℘( x | ω1, ω2 ) = 2 + 2 − 2 , (2.12) x (x + 2mω1 + 2nω2) (2mω1 + 2nω2) {m,n}6={0,0}

with primitive periods 2ω1 and 2ω2. More generally the interparticle potentials of Calogero- Moser systems are constructed from root systems of Lie algebras [17]. So for other types of CM systems the interparticle potential is not simply V (qi − qk), but rather some more complicated function based on the functions V (x). For example, the DN type system keeps the interparticle potential [17]

2 U(qi, qj) = g V (qi − qj) + V (qi + qj) . (2.13) A more complete classiﬁcation of these root system Calogero-Moser systems is provided by Perelomov in [17]. There are also relativistic, quantum, and relativistic quantum Calogero-Moser systems. For example, the nonrelativistic quantum AN−1 type system (without external potential) is written [13]

N N 1 X 2 X0 Hˆ = − i ∂ + g(g − ) V (q − q ). (2.14) 2 ~ xj ~ j k j jk For more on relativistic and quantum CM systems, see e.g. Ruijsenaars in [13]. Needless to say, the theory of Calogero-Moser systems is rich and extensive, while our time is ﬁnite. Thus, in the rest of this work we restrict ourselves to the classical Calogero Moser systems of AN−1 type with external potential, as deﬁned by (2.11). These systems will in the following be referred to simply as the Calogero-Moser systems.

Note that pole at x = 0 of the interparticle potential V (x) implies a divergence for vanishing interparticle separation: qi − qj → 0 ⇒ V (qi − qj) → ∞. Hence a CM system with repulsive interaction, modelled by g2 > 0, aﬀords a stable system with particle trajectories that cannot intersect (so the ordering of particles in one dimension is conserved). For the rest of this work we assume g2 > 0, such that our CM systems are composed of mutually repulsive particles.

Chapter 2 8 Section 2.2 A. R. Philip Soliton solutions to Calogero-Moser systems

2 Remark. The two most prominent CM models are given by V (qi − qj) = 1/(qi − qj) , W (x) = 2 2 2 2ω qi with qi ∈ R, and V (qi − qj) = 1/ sin (qi − qj) with qi ∈ [0, 2π)/2πZ (so the qi represent angles on the the unit circle). More generally, qi ∈ M where M is some smooth connected 1-manifold, referred to as the conﬁguration manifold of the Hamiltonian system. The set of possible M then comprises all spaces topologically isomorphic to either the line or the circle. One beneﬁt with the two models above is that they are constrained, whereas the model 2 V (qi − qj) = 1/(qi − qj) , W (qi) = 0 with qi ∈ R, for example, yields a system of particles that drift apart with time.

In the absence of an external potential, i.e. W (x) = 0, the systems included in the Hamiltonian (2.11) are all Liouville integrable, in that they possess (a minimum of) N distinct conserved quantities, H1, ..., HN , in involution:

{Hi,Hj} = 0, where { , } here denote Poisson brackets. Out of convenience in the following we refer to these systems as the isolated Calogero-Moser systems, as their particles interact only with each other, and are free from external inﬂuences. We may then state that the isolated CM systems are Liouville integrable. This can be proven, for example, by using their Lax representations. The integrability of the elliptic class IV system (which is also the most general of the isolated systems included in (2.11)) has been proven by Perelomov in [14], see also [17]. Alternatively, for the isolated systems of type I, II and III it follows trivially from the Kazhdan-Kostant-Sternberg construction [16] (as Tr{Y n} : n = 1, ..., N yields N involute integrals of motion, see §2.3). With an arbitrary external potential in (2.11) the resulting system would generally not be integrable. However, it is known that the class I interparticle potential combined with a harmonic external potential, W (x) = 2ωx2, yields an integrable system. For convenience we classify this system, i.e.

N N 0 1 X 2 2 2 2 X 1 2 + H + = p + 2ω q + g , (I ) I 2 i i q − q i=1 i,j=1 i j

+ as the I Calogero-Moser system. Beyond the AN−1 type Hamiltonian (2.11) a few more integrable CM systems with external potential are known. For the quantum case a few can be found in table 1 of [15].

Lastly, the Weierstrass function is only determined as an ordinary function of x after its two primitive periods 2ω1, 2ω2 have been speciﬁed, whence diﬀerent combinations of (ω1, ω2) in ℘(x), correspond to distinct potentials V (x). So the class IV system comprises a number of inequivalent CM systems. In fact, as evidenced by the following limits of the Weierstrass function [12]

2 ω1 → ∞, ω2 → i∞ : ℘(x) → 1/x , (2.15) 2 2 2 ω1 → ∞, ω2 = iπ/(2a): ℘(x) → a /3 + a / sinh ax, (2.16) 2 2 2 ω1 = π/(2a), ω2 → i∞ : ℘(x) → −a /3 + a / sin ax, (2.17)

Chapter 2 9 Section 2.2 A. R. Philip Soliton solutions to Calogero-Moser systems

wherein the V (x) of classes I, II and III are recovered, it comprises all potentials permitted by (2.11). Thus (2.11) can be written more compactly as

N 1 X X0 H = p2 + W (q ) + g2 ℘(q − q ), (2.18) CM 2 j j j k j jk which up to scaling corresponds to the most general AN−1 type Calogero Moser system known [11] [12].

Chapter 2 10 Section 2.2 A. R. Philip Soliton solutions to Calogero-Moser systems

2.3 The Kazhdan-Kostant-Sternberg (KKS) construction of CM systems

Our presentation in the following is based on P. Etingof in [16] and Perelomov in [17].

On Hamiltonian reduction. The presence of continuous symmetries within a Hamiltonian system (T ∗M, H) reduces its effective degrees of freedom in that any initial configuration ( q(0), p(0) ) ∈ T ∗M de- ∗ fines a submanifold T Mred of lower dimension, within which the continued evolution of the system is constrained. So for a pair of initial q(0), q˜(0) related by a symmetry transformation T , such that q˜(0) = T q(0), it holds in continuation that q˜(t) = T q(t). Upon repeated transformations along a symmetry ’direction’, we pass through redundant descriptions of the same physical configuration of the system. This redundancy can be removed if the Hamiltonian system is explicitly reformulated as a system of fewer degrees of freedom, by the procedure of Hamiltonian reduction.

In essence, Hamiltonian reduction amounts to expressing the Hamiltonian system in terms of one of its degenerate conﬁgurations, after which the reduced symmetry becomes ’hidden’1 and the Hamiltonian function itself typically becomes more complicated. First however, the reduced manifolds must be determined. To this end consider a symmetry group G with Lie algebra g, acting on the symplectic phase manifold T ∗M of the Hamiltonian system. To the coadjoint algebra g∗ there is associated a map µ(x): T ∗M → g∗, called the moment map, satisfying certain properties (see [17] for details). The moment map is invariant under the action of G. Hence, if G is a symmetry of the Hamiltonian function, µ(x) generates the conserved quantities c ∈ g∗ associated with the symmetry. If the moment map can be found, its reverse for ﬁxed values c, µ−1(c), yields the submanifolds related to conserved quantity c, and its quotient with respect to the −1 c-invariant subgroup Gc, µ (c)/Gc, yields the reduced phase manifold.

KKS construction of the CM systems. That the reduced Hamiltonian typically is more complicated than the original one, suggests in reverse a use of Hamiltonian reduction to construct complicated Hamiltonian systems from simpler ones, with more degrees of freedom, and where the ’hidden’ symmetries of the complicated system are apparent. The isolated CM systems of class I, II and III afford just such a construction, from Hamiltonian functions on the larger configu- 0 ration space XN = {X} of N × N traceless matrices X, known as the Kashdan, Kostant & Sternberg (KKS) construction. ∗ 0 In the KKS construction, the phase manifold is the tangent bundle T XN , which might itself be mapped by canonical pairs of traceless N × N matrices (X,Y ) (since covectors are identified with vectors by the inner product Tr{XY }), and the Hamiltonian functions are invariant with respect to similarity transformations (X,Y ) → (UXU −1,UYU −1), (2.19) where U is an invertible N × N matrix. These similarity transformations constitute the action of the symmetry group P GLN (C) with Lie algebra g = slN (C) [16]. The action of

1That is, not apparent prima facie, though still technically present.

Chapter 2 11 Section 2.3 A. R. Philip Soliton solutions to Calogero-Moser systems

∗ 0 the group induces a vector ﬁeld on T XN , which equivalently could be generated by the Hamiltonian function [17] F = Tr{ξ[X,Y ]}, ξ ∈ g. (2.20) The above must constitute the inner product between a vector ξ and a covector [X,Y ]. Thus a moment map µ : M → g∗, associated with the symmetry, is given by the commutator µ = [X,Y ]. (2.21) Then the set of reduced phase manifolds along the symmetry direction c, comprises (X,Y ) such that µ = [X,Y ] = c, (2.22) with the general solution [17] X = UQU −1,Y = ULU −1, (2.23) where   Qnm = δnmqn(t), (2.24) ig  Lnm = δnmpn(t) + 1 − δnm . qn − qm Remark. The L might be recognized as the Lax matrix of the rational Calogero Moser system, see §2.4.2.

I: Rational CM system The rational CM system derives from the Hamiltonian function 1 H = TrY 2 , (X,Y ) ∈ T ∗X0 , (2.25) 2 N 0 describing free motion on XN . Due to the cyclic property of the trace operation, the Hamiltonian must be invariant with respect to the group of similarity transformations. Consequently, the system may be reduced by projecting for instance X → Q, Y → L, and that this yields the rational CM system is demonstrated by the fact that

1 1 X X0 −2 H = TrL2 = p2 + g2 q − q . (2.26) 2 2 i i j i ij Remark. Tr{Y n} : n = 1, .., N yields N integrals of motion in involution. Hence the N-dimensional reduced system must be integrable.

I+: rCM in external harmonic potential Including an external harmonic potential, the isolated rCM Hamiltonian is recast into 0 1 X 2 2 2 X g 2 H + = p + ω q + . (2.27) I 2 i i q − q i ij i j 0 The system derives from harmonic motion on XN 1 H = TrY 2 + ω2X2 , (X,Y ) ∈ T ∗X0 , (2.28) 2 N through the same projection X → Q, Y → L, with X and L as in the previous.

Chapter 2 12 Section 2.3 A. R. Philip Soliton solutions to Calogero-Moser systems

II: Hyperbolic CM system

0 The hyperbolic CM system derives from motion along hyperbolic geodesics on XN , expressed by the Hamiltonian function [16] 1 H = Tr XY 2, (X,Y ) ∈ T ∗X0 . (2.29) 2 N The projection in this case yields the reduced system

N 1 X X0 qiqj H = (q p )2 + g2 . (2.30) tCM 2 i i (q − q )2 i ij i j

However, by a canonical transformation to hyperbolic coordinates 2aqˆi = ln qi, pˆ = qipi it follows that

N 1 X X0 1 H = pˆ2 + g2a2 , (2.31) tCM 2 i sinh2{a(ˆq − qˆ )} j ij i j

and that

iga Lnm = δnmpˆn + 1 − δnm . (2.32) sinh {a(ˆqn − qˆm)}

The analog transformation 2aXˆ = ln X, Yˆ = XY implies in turn 1 H = Tr Yˆ 2, (2.33) 2

from which the reduction is immediately accomplished by the projection Xˆ → Q, Yˆ → L, with   Qnm = δnmqˆn(t), (2.34) iga  Lnm = δnmpˆn + 1 − δnm . sinh {a(ˆqn − qˆm)} The L may now be recognized as a Lax matrix of the hyperbolic type II CM system, see §2.4.2.

III: trigonometric CM system The KKS procedure for the trigonometric CM system:

N 1 X X0 1 H = pˆ2 + g2a2 , (2.35) tCM 2 i sin2{a(ˆq − qˆ )} j ij i j

follows directly from the hyperbolic case in replacing qˆj → iqˆj.

Chapter 2 13 Section 2.3 A. R. Philip Soliton solutions to Calogero-Moser systems

2.4 Lax representations

Lax representation theory proceeds from the fact, noted by P. D. Lax (see [26]), that the eigenvalue λ to a linear operator L(s) whose evolution is determined by another linear operator M according to dL = [L, M], (2.36) ds is invariant to spectral deformations in the parameter s: dλ/ds = 0. The L by convention is termed a Lax operator and the M an auxiliary operator. Together, a pair of operators (L, M) satisfying (2.36) is said to constitute a Lax pair [29]. For a dynamical system the Lax representation is constituted by some particular Lax pair, for which (2.36) becomes equivalent to the dynamical equation(s) of the system, and the eigenvalues of the Lax operator become equivalent to its integrals of motion (not necessarily all of them). The Lax representations, thus, oﬀer an equivalent way of writing the dynamical equations of a system, in a form that incorporates the integrals of motion directly. This fact has to great eﬀect been used to prove the integrability of a number of dynamical systems.

2.4.1 Lax representations of many-body Hamiltonian systems For a many-body Hamiltonian system

H(q1, ...qN , p1, ...pN ), (2.37)

a Lax representation is given by a Lax pair of N × N matrices L(q1, ...qN , p1, ...pN ) and M(q1, ...qN , p1, ...pN ), for which the Lax evolution equation in the time parameter, t

L˙ = [L, M], (2.38)

becomes equivalent to Hamilton’s equations of motion: ∂H ∂H q˙n = , p˙n = − . (2.39) ∂pn ∂qn

In consequence, the N eigenvalues λn of a Lax matrix L, are conserved quantities/integrals of motion of the system H.

Remark. The time-dependence of the Lax pair is implicit in qn(t) and pn(t).

If one could also show that the λn are distinct, and in involution:

{λi, λj} = 0, (2.40) where { , } denotes Poisson commutation, it would follow that the system H is Liouville integrable, with its N integrals of motion aﬀorded by the eigenvalues λn. Hence, a Lax representation of a Hamiltonian system has the capacity to immediately generate the integrals of motion (or action variables) of the system. If the system is integrable with known integrals of motion, its Lax representation can be constructed even more explicitly, see appendix A.1. However, the Lax representations of the Calogero-Moser systems were originally found by Calogero from an ansatz, see appendix A.2.

Chapter 2 14 Section 2.4 A. R. Philip Soliton solutions to Calogero-Moser systems

Equivalent Lax representations I Lax representations with identical eigenvalues of L are equivalent. Hence the similarity transformation L˜ = ULU −1, yields a new Lax matrix to the system, satisfying its own Lax evolution equation L˜˙ = [L,˜ M˜ ] with some auxiliary matrix M˜ . It follows that h i L˙ − [L, M] = U L˜˙ − L,˜ M˜ U −1, (2.41)

which can be expanded into h i ULU˜˙ −1 + U˙ LU˜ −1 − ULU˜ −1UU˙ −1 − [L, M] = ULU˜˙ −1 − ULU˜ −1,UMU˜ −1 , (2.42)

wherefrom h i [L, M] = L, UMU˜ −1 − UU˙ −1 . (2.43)

Therefrom we conclude that M transforms according to M = UMU˜ −1 −UU˙ −1. Including the freedom inherent to the commutator operation, the following Lax pair equivalence relations are obtained ( L = ULU˜ −1 + c1, (2.44) M = UM˜ + f(t)L˜U −1 − UU˙ −1 + g(t)1.

2.4.2 Lax representations of isolated CM systems The isolated Calogero-Moser systems

N 1 X X0 H = p2 + g2 ℘(q − q ), (2.45) CM 2 j j k j jk

are compatible with Lax pairs on the form   L = δ p + (1 − δ )α(q − q ),  nm nm n nm n m N (2.46) X  M = δ β(q − q ) + (1 − δ )α0(q − q ),  nm nm n l nm n m  l6=n

in case certain requisites on the functions α(x) and β(x) are met. These requisites can be determined in two consecutive operations:

1. Be the Lax evolution equation L˙ = [L, M] expanded by (2.46), Hamilton’s eqs. of motion are recovered should it hold that V (x) = α(x)α(−x), β(−x) = β(x) and

α(x)α0(y) + α(y)α0(x) = α(x + y) β(x) − β(y) , (2.47)

according to calculations performed in appendix A.2. Equation (2.47) is conventionally termed the Lax functional equation, and may be considered as the basis of a Lax functional representation furnished by the Lax pair of functions α(x), β(x) . In analogy with the Lax matrices, we will refer to α(x) as a Lax function and β(x) an auxiliary function.

Chapter 2 15 Section 2.4 A. R. Philip Soliton solutions to Calogero-Moser systems

2. A Taylor expansion of (2.47) about y = −x + , → 0 yields, on grounds of consistency, that the asymptotic behaviour of α(x) about x = 0 necessarily must be of the kind c α() = −1 + c + c + ... + c 2k−1 + ..., c 6= 0, k ∈ Z+, (2.48) 0 1 2k−1 −1 and further that β(x) = −V (x)/c−1 + β0 = −α(x)α(−x)/c−1 + β0 with β0 constant, according to calculations performed in appendix A.3. Remark. It may alternatively be derived that β(x) = α00(x)/[2α(x)], see [17].

We conclude that the Lax pair of a CM system may be determined from an appropriate Lax function, whose properties we summarize in the deﬁnition:

Deﬁnition (Lax function): A function α(x) is a Lax function to an isolated CM system with interparticle potential V (x) = α(x)α(−x) if α() → c−1/ as → 0 and the functional equation (2.47) is satisﬁed for β(x) = −V (x)/c−1 + β0.

Equivalent Lax representations II As with the Lax operator pair, the Lax function pair is not uniquely determined. In fact, the functional equation is invariant under the combined transformations [11] ( α(x) → α˜(x) = becxα(ax), (2.49a) ˜ β(x) → β(x) = abβ(ax) + β0, (2.49b) whereupon the interparticle potential itself transforms in the simple manner

V (x) → V˜ (x) = b2V (ax). (2.49c)

Although by no means obvious, the equivalence transformations when applied to the Taylor expansion of the Lax functional equation, imply that for any CM potential V (x) it is possible to choose an odd α(x): α(−x) = −α(x). Consult [11] or appendix A.3.

The special Lax function and special functional equation To the special (or degenerate) CM systems of class I, II and III, there exists an odd Lax 0 function with the additional property, that β(x) = −α (x) + β0. A Lax function of this kind we term a special Lax function. Remark. In order for β(x) to be an even function above, α(x) must be odd. For a special lax function, the functional equation reduces to

α(x)α0(y) − α(y)α0(x) = α(x + y) α0(y) − α0(x) , (2.50) which we henceforth refer to as the special functional equation. We conclude that the following is a Calogero-Moser Lax pair   L = δ p + (1 − δ )α(q − q ),  nm nm n nm n m N (2.51) X  M = −δ α0(q − q ) + (1 − δ )α0(q − q ),  nm nm n l nm n m  l6=n

Chapter 2 16 Section 2.4 A. R. Philip Soliton solutions to Calogero-Moser systems

which is determined by a special Lax function α(x), whose properties we summarize in the deﬁnition:

Deﬁnition (Special Lax function). An odd function α(x) is a special Lax function to the isolated CM system with interparticle potential V (x) = α(x)α(−x) if α() = c−1/ 0 2 as → 0, α (x) = −α (x)/c−1 + const., and if it satisﬁes the special functional equation (2.50).

2.4.3 Examples of Calogero-Moser Lax representations I: Rational CM (rCM) system Provided a Lax function pair α(x) = i/x, β(x) = i/x2, the isolated CM system with V (x) = 1/x2 can equivalently be expressed in terms of the Lax pair   ig  Lnm = δnmpn + 1 − δnm ,  qn − qm (2.52)  X ig ig  Mnm = δnm − 1 − δnm .  2 2 l6=n qn − ql qn − qm

Note that the Lax function α(x) is special, virtue of β(x) = −α0(x), and satisﬁes the special functional equation (2.50).

II: Hyperbolic CM (hCM) system For the isolated CM system with V (x) = a2/ sinh2 ax, one might choose α(x) = iga/ sinh (ax), β(x) = iga2/ sinh2 ax, whereby   iga  Lnm = δnmpn + 1 − δnm ,  sinh {a(qn − qm)} (2.53) 2 2  X iga iga cosh{a(qn − qm)}  Mnm = δnm − 1 − δnm .  sinh2{a(q − q )} sinh2{a(q − q )}  l6=n n l n m

2 2 d However, another choice is α(x) = iga coth (ax), β(x) = iga / sinh ax, where dx coth x = −1/ sinh2 x implies β(x) = −α0(x) ⇒ (2.50) is satisﬁed. So this α(x) is a special Lax function, with corresponding Lax pair   L = δ p + 1 − δ iga coth {a(q − q )},  nm nm n nm n m  (2.54) X iga2 iga2  M = δ − 1 − δ .  nm nm sinh2{a(q − q )} nm sinh2{a(q − q )}  l6=n n l n m

Chapter 2 17 Section 2.4 A. R. Philip Soliton solutions to Calogero-Moser systems

III: Trigonometric CM (tCM) system For the trigonometric system, with V (ax) = a2/ sin2 ax, one might choose α(x) = iga/ sin (ax), β(x) = iga2/ sin2 ax whereby   iga  Lnm = δnmpn + 1 − δnm ,  sin {a(qn − qm)} (2.55) 2 2  X iga iga cos{a(qn − qm)}  Mnm = δnm − 1 − δnm ,  sin2{a(q − q )} sin2{a(q − q )}  l6=n n l n m

2 2 d 2 but also α(x) = iga cot (ax), β(x) = iga / sin ax, where dx cot x = −1/ sin x implies that β(x) = −α0(x) ⇒ the special functional equation (2.50) is satisﬁed, and V (x) = α0(x) + const.. The corresponding Lax pair is written   L = δ p + 1 − δ iga cot {a(q − q )},  nm nm n nm n m  (2.56) X iga2 iga2  M = δ − 1 − δ .  nm nm sin2{a(q − q )} nm sin2{a(q − q )}  l6=n n l n m

IV: General elliptic CM system The general form of α(x) has been determined to [30]

σ(x − α) α(x) = exp{ζ(α)x}, (2.57) σ(α)σ(x)

where σ and ζ are, respectively, the elliptic sigma and zeta Weierstrass functions. If α(x) is taken odd we have the following set of possible solutions

cn(ax) dn(ax) 1 α(x) = a , a , a , (2.58) sn(ax) sn(ax) sn(ax)

in terms of the Jacobi elliptic functions. It holds that [11]

√ σ(x) σ1(x) σ2(x) sn(x) = e1 − e3 , cn(x) = , dn(x) = , (2.59) σ3(x) σ3(x) σ3(x)

where σj(x) = exp{−ηjx}σ(x + wj)/σ(wj). See appendix B of [11] for clariﬁcation of notation.

Remark. One should be aware that the special functional equation is not applicable to the general CM system (type IV): though the Weierstrass zeta function ζ(x) is deﬁned by ℘(x) = −ζ0(x) it does not satisfy (2.50) but instead

α(x)α0(y) − α(y)α0(x) = α(x + y) α0(y) − α0(x) + Γ(x) − Γ(y), (2.60)

0 1 00 where Γ(x) = α(x)α (x) + 2 α (x) [7]. And ζ(z) anyway cannot be a Lax function, since ζ(x)ζ(−x) 6= ℘(x). See discussion in §3.3.2.

Chapter 2 18 Section 2.4 A. R. Philip Soliton solutions to Calogero-Moser systems

2.5 Olshanetsky Perelomov (OP) Projection method

As mentioned in §2.2 the isolated Calogero-Moser systems are Liouville integrable, since they possess N distinct integrals of motion in involution. In principle they could hence be solved by repeated application of Hamiltonian reduction. Though, for these particular systems this algorithm is not applicable in practice. However, an alternative method of integration, applicable to the isolated CM systems of class I, II and III, as well as to the I+ system, has been developed by Olshanetsky and Perelomov [20], [21]. Their projection method amounts to the integration of an auxiliary Hamiltonian system, from which the solutions to the considered system can be recovered by projection. That is, the solutions q(t) ∈ M to a system (T ∗M, H) are obtained from the integration of an auxiliary system (T ∗M˜ , H˜) through a map π : M˜ → M that is a projection: q(t) = πq˜(t).

Remark. For consistency π in addition should satisfy the condition that, if πq˜1(0) = πq˜2(0), then πq˜1(t) = πq˜2(t) (see [17]). A projection of this kind is facilitated by Hamiltonian reduction, with the beneﬁt that the auxiliary system typically would keep a simpler Hamiltonian function, and thus likely be easier to solve. In the following we refer to the KKS construction of the CM 0 systems (see §2.3) from Hamiltonian reduction of systems on the vector space XN = {X} of N × N traceless matrices X. We recommend for §2.3 to be read prior to or in tandem with the following.

2.5.1 I: Rational CM system

0 Free motion on the vector space XN = {X} of N × N traceless matrices X 1 H = TrY 2 , (X,Y ) ∈ T ∗X0 , (2.61) 2 N reduces into the isolated rCM system in projecting X → Q, Y → L where   Qnm = δnmqn(t), (2.62) ig  Lnm = δnmpn(t) + 1 − δnm . qn − qm

In addition, the equation of motion to (2.61), X¨ = 0, integrates directly to

0 X = At + B, A, B ∈ XN . (2.63)

Now consider a similarity transformation

X(t) = U(t)Q(t)U −1(t), (2.64) which by time diﬀerentiation yields

X˙ = A = UQU˙ −1 + UQU˙ −1 − UQU −1UU˙ −1 (2.65) = UQU˙ −1 + UU −1UQU˙ −1 − UQU −1UU˙ −1 (2.66) = UQ˙ − [Q, M]U −1, (2.67)

Chapter 2 19 Section 2.5 A. R. Philip Soliton solutions to Calogero-Moser systems where M = U −1U˙ . Writing L = Q˙ −[Q, M] it is similarly obtained, from the second time derivative, that

X¨ = U L˙ − [L, M] U −1 = 0, (2.68) which implies that L and M form a Lax matrix pair (see §2.4.2). Given an M, the U(t) is deﬁned up to initial value by U˙ = UM. Setting U(0) = 1 it −1 follows that A = U(0)L(0)U (0) = L(0) and B = X(0) = Q(0). Thus the qn(t) may be recovered algebraically as the eigenvalues of

X(t) = L(0)t + Q(0), (2.69) where t Xnm(t) = δnm qn(0) + pn(0)t + 1 − δnm ig . (2.70) qn(0) − qm(0)

2.5.2 I+: Rational CM system with external harmonic potential

0 For harmonic motion on XN : 1 H = TrY 2 + ωX2 , (X,Y ) ∈ T ∗X0 , (2.71) 2 N the equation of motion X¨ + ω2X = 0 integrates to A X = sin ωt + B cos ωt, A, B ∈ X0 . (2.72) ω N From the similarity transformation, with U(0) = 1

X(t) = U(t)Q(t)U −1(t), (2.73) it follows that B = X(0) = Q(0); and from

X˙ (t) = U(t)L(t)U −1(t), (2.74)

˙ that A = X(0) = L(0). The qn(t) of the reduced system, are thus recovered as the eigenvalues of sin ωt ig sin ωt Xnm(t) = δnm qn(0) cos ωt + pn(0) + 1 − δnm . (2.75) ω ω qn(0) − qm(0)

Note however, that the L and M are not proper Lax matrices to the reduced system, but satisfy

L˙ − [L, M] = −ω2X, (2.76) whence also the eigenvalues of L are not conserved quantities of the reduced system.

Chapter 2 20 Section 2.5 A. R. Philip Soliton solutions to Calogero-Moser systems

2.5.3 II: Hyperbolic CM system

0 We now wish to consider hyperbolic motion on the space XN . The Hamiltonian in this case is written 1 H = Tr XY 2, (X,Y ) ∈ T ∗X0 . (2.77) 2 N which by a canonical transformation to the hyperbolic set of canonical coordinates aXˆ = (1/2) ln X, Yˆ = XY , becomes 1 H = Tr Yˆ 2. (2.78) 2 From §2.3 we know that the latter reduces into the hyperbolic CM system in projecting Xˆ → Q, Yˆ → L, where   Qnm = δnmqˆn(t), (2.79) iga  Lnm = δnmpˆn + 1 − δnm , sinh {a(ˆqn − qˆm)}

where it holds that aqˆi = (1/2) ln qi, pˆi = qipi. In hyperbolic coordinates we have that

Xˆ(t) = L(0)t + Q(0), (2.80) which, in the original coordinates may be written

X(t) = exp{aL(0)t} exp{2aQ(0)} exp{aL(0)t}. (2.81)

2.5.4 III: Trigonometric CM system The trigonometric case is obtained directly from the hyperbolic, in appropriate insertion of factors i. Hence the eigenvalues of

X(t) = exp{iaL(0)t} exp{2iaQ(0)} exp{iaL(0)t}, (2.82)

yields the mapping of the real solutions qˆn(t) to the complex unit circle qn(t) = exp{2iaqˆn(t)}.

Chapter 2 21 Section 2.5 A. R. Philip Soliton solutions to Calogero-Moser systems

2.6 Bäcklund transformations

In general terms, a Bäcklund transformation (BT) between a pair of partial- (and in general nonlinear-) diﬀerential equations

P [ u ] = 0 and Q[ v ] = 0, (2.83) maps solutions of one to solutions of the other: u → v and/or v → u. The virtue of the transformation should be self evident, as it might for instance render unknown solutions, to a particularly complicated P [ u ] = 0, determinable from familiar solutions of some simpler Q[ v ] = 0. However the possibilities to this end should not be exaggerated. For one, the Bäcklund transformations typically assert limitations upon the solutions it can map between, and thence does not, in general, serve to connect all solutions u, v to its partial diﬀerential equations. They are also non-trivial to construct, and no general construction method is known, although a number of such methods are available for various special cases.

2.6.1 General discussion The Bäcklund transformations often appear as sets of coupled relations (mixed in u and v) that together imply P [ u ] = 0 and Q[ v ] = 0 at the same time. This applies for example to ( uv + 2cv = 0, x (2.84) 2 cuxv − u v/2 + cvt = 0, which constitutes a BT between the Burger’s equation, ut + uux − cuxx = 0, and the heat equation, vt = vxx; and to the BT ( √ u + v = 2 exp{(u − v)/2}, x x √ (2.85) ut − vt = 2 exp{(u + v)/2}, that connects Liouville’s equation, uxt = exp{u}, with the simple equation vxt = 0. In contrast, the Miura transformation

2 u = v + vx, (2.86) implies the KdV equation, ut − 6uux + uxxx = 0, if v solves the modiﬁed KdV equation

2 vt − 6v vx + vxxx = 0, (2.87) but technically does not imply the mKdV (unless u satisfies the KdV equation). A few authors define the BTs to comprise exactly a pair of relations. However, this would seem to conflict with the many-body/lattice Hamiltonian Bäcklund transformations, such as the Kac-Van Morbecke system of equations [22]

x˙ n = exp{xn − xn+1} + exp{xn−1 − xn}, n = 1, ..., N, x0 = −∞, xN+1 = ∞ (2.88) which serves to connect a pair of disjoint Toda lattices: x¨n = exp{xn−2 − xn} − exp{xn − xn+2}.

Chapter 2 22 Section 2.6 A. R. Philip Soliton solutions to Calogero-Moser systems

Auto-Bäcklund transformations An auto-Bäcklund transformation (aBT) is a mapping between solutions to the same partial diﬀerential equation, so we would have P [ u ] = 0 ≡ Q[ u ] = 0. They commonly recur in the context of soliton theory, as a means to generate solutions of multiple solitons from ’primitive solutions’ of just one or two solitons. Prominent examples of aBTs include the Cauchy-Riemann equations ( u − v = 0, x y (2.89) uy + vx = 0,

implying the Laplace equation, wxx + wyy = 0, at once for w = u and w = v; and the relations ( (u + v) = 2a sin {(u − v)/2}, x (2.90) (u − v)t = (2/a) sin {(u + v)/2},

that reproduce the sine-Gordon equation, wxt = sin w, in w = u and w = v. One should be aware that the term Bäcklund transformations in much of the literature refers to the auto-Bäcklund transformations specifically. The aBTs between continuous nonlinear systems, are intimately related to their Lax representations and to the the inverse scattering transform. The existence of a Lax representation is a necessary condition for the inverse scattering transform to be applicable [27]. If a Lax representation exists, it may be used to define a connection on a certain principal fibre bundle, from which the aBTs may be recovered as automorphisms of the connection (see [27]). Thus, aBTs can sometimes be constructed through methods of differential geometry.

Hamiltonian Bäcklund transformations A Bäcklund transformation relating a pair of Hamiltonian systems, maps between their canonical coordinates, (x(t), x˙(t)) and (z(t), z˙(t)) respectively. An example is the Kac- Van Moerbeke system of equations, (2.88), as given previously, that relates a pair of disjoint Toda lattices. In consequence, the Hamiltonian auto-Bäcklund transformations must correspond to canonical transformations, i.e. they preserve the form of Hamilton’s equations. Actually they belong to subset of canonical transformations, that in addition preserve the Hamiltonian itself, as otherwise the conserved quantities of the system would be aﬀected by the transformation [28]. Auto-Bäcklund transformations to Hamiltonian systems, are intimately related to their Lax representations (treated in §2.4.2). If the considered Hamiltonian is integrable with a Lax representation in a Lax pair, (L, M), the integrals of motion may be recovered as the coeﬃcients of the characteristic polynomial

det v − L. (2.91)

Then the invariance of the coeﬃcients under an auto-Bäcklund transformation, must be equivalent to the invariance of the spectrum of L, which requires that L is indeed a Lax operator [28]. This is one way to demonstrate the equivalence between Hamiltonian aBTs and their Lax representations.

Chapter 2 23 Section 2.6 A. R. Philip Soliton solutions to Calogero-Moser systems

2.6.2 Bäcklund transformations of Calogero-Moser systems Bäcklund transformations between solutions of the Calogero-Moser (CM) systems were introduced by S. Rauch Wojciechowski [22] in 1982. These results already encompass the isolated CM systems as well as the class I+ system. In the following, we present the same BTs, between systems of class I, II, III, and I+, but in a manner that emphasises the role of their special Lax functions. The relevant proof can be found in appendix B.1. This proof adds some novelty, as it is based solely on the properties of the special Lax functions, something that we have not hitherto encountered in the literature.

First, recall the properties of what we term a special Lax function (see §2.4.2),

Deﬁnition (Special Lax function). An odd function α(x) is a special Lax function of the CM system with interparticle potential V (x) = α(x)α(−x) if α() = c−1/ as → 0, 0 2 α (x) = −α (x)/c−1 + const., and if it satisﬁes the special functional equation

α(x)α0(y) − α(y)α0(x) = α(x + y) α0(y) − α0(x) . (2.50)

and consider N complex canonical coordinates xi, x˙ i ∈ C, and M canonical coordinates + zn, z˙n ∈ C. The Bäcklund transformations between CM systems of type I, II, III and I can then be written on the form  N M  X X  x˙ i = − α(xi − xj) + α(xi − zn) + λ + iωxi,   j6=i n (2.92) M N  X X  z˙ = α(z − z ) − α(z − x ) + λ + iωz .  n n m n k n  m6=n k with α(x) the special Lax function corresponding to V (x), λ constant, and ω = 0 for all but class I+. The system (2.92), implies the pair of uncoupled dynamical equations

X 0 2 X 0 2 x¨i = c V (xi − xk) − ω xi, z¨n = c V (zn − zm) − ω zn, (2.93) k6=i m6=n a fact of which we provide an elegant proof in appendix B.1. Thus {xi} and {zn} belong to separate CM systems of interparticle potential V (x). Note that, although the coordinates and momenta of either system may be taken real, the coordinates or the momenta of the other system then must necessarily be complex. Also, for some calculations it might be of interest to note the following equivalent expressions

0 2 0 00 V (x) = −∂x α (x) = −2α(x)α (x) = c−1α (x). (2.94)

Equivalence to Lax representations Wojciechowski in [22] mentions an additional fact regarding the CM Bäcklund transformations. For convenience we restrict ourselves here to the rational CM system, where α(x) ∝ 1/x (see [22] for more general results). It holds that the rational Bäcklund transformations may be derived from solutions

M Πn α(z − zn) iε(λz−λ2t) ψ(z, t) = N e , (2.95) Πi α(z − xi)

Chapter 2 24 Section 2.6 A. R. Philip Soliton solutions to Calogero-Moser systems

to the Schrödinger equation, iψt = −ψxx + U(z)ψ, as the residues in z → xi and z → zn respectively. We perform the calculation in appendix B.2. Furthermore, when N = M the solutions may be decomposed into

M iε(λz−λ2t) X ψ(z, t) = e 1 + kiα(z − zi) , i whereafter the residue calculation instead yields the Lax representations of the Calogero Moser systems [22]. This explicitly demonstrates for the rational case that the CM auto- Bäcklund transformations are equivalent to the CM Lax representations.

2.6.3 Examples of BTs for CM systems Below we write out explicitly the Bäcklund transformations for the isolated Calogero Moser systems of class I, II, and III as well as for the class I+ system. The Bäcklund transformations for the general elliptic system is given in [22] or below in §3.3.2, and a proof can be found in [31]

I: rational CM system The special Lax function α(x) = ig/x, and ω = 0, results in the BT

N M X 1 X 1 x˙ j = − ig + ig , (2.96) xj − xk xj − zn k6=j n N M X 1 X 1 z˙n = ig − ig , (2.97) zn − zm zn − xl m6=n l

which upon time diﬀerentiation results in the uncoupled dynamical equations

N M 2 X 1 3 2 X 1 3 x¨j = 2g , and z¨n = 2g . (2.98) xj − xk zn − zm k6=j m6=n

These correspond to Newton’s eqs. of motion to the rational CM systems

Z Z 0 1 X 2 1 X g 2 HCM [Z] = w˙ j + , (2.99) 2 2 wj − wk j jk

in wi = xi, Z = N and, respectively, wi = zi, Z = M.

I+: rCM with harmonic ext. potential With ω 6= 0 the BT above modiﬁes to

N M X 1 X 1 x˙ j = − ig + ig + iωxj, (2.100) xj − xk xj − zn k6=j n N M X 1 X 1 z˙n = ig − ig + iωzj, (2.101) zn − zm zn − xl m6=n l

Chapter 2 25 Section 2.6 A. R. Philip Soliton solutions to Calogero-Moser systems

which implies N 2 X 1 3 2 w¨j = 2g − ω wj, (2.102) wj − wk k6=j

for wi = xi and wi = zi respectively. This corresponds to Newton’s equations for the rational CM system with harmonic external potential:

N N 0 1 X 2 2 1 X g 2 HCM [N] = w˙ j + ω wj + . (2.103) 2 2 wj − wk j jk

II: hyperbolic CM system For the hyperbolic system N 1 X X0 1 H = p2 + g2a2 , (2.104) hCM 2 i sinh2{a(q − q )} j ij i j the special Lax function α(x) = iga coth x yields the Bäcklund transformations N M X X x˙ j = − iga coth{a(xj − xk)} + iga coth{a(xj − zn)}, (2.105) k6=j n N M X X z˙n = iga coth{a(zn − zm)} − iga coth{a(zn − xl)}, (2.106) m6=n l which imply N X cosh{a(wj − wj)} w¨ = 2g2a3 , (2.107) j sinh3{a(w − w )} k6=j j k in wi = xi and wi = zi.

III: trigonometric CM system For the trigonometric system N 1 X X0 1 H = p2 + g2a2 , (2.108) tCM 2 i sin2{a(q − q )} j ij i j the special Lax function α(x) = iga cot ax yields the BTs N M X X x˙ j = − iga cot{a(xj − xk)} + iga cot{a(xj − zn)}, (2.109) k6=j n N M X X z˙n = iga cot{a(zn − zm)} − iga cot{a(zn − xl)}. (2.110) m6=n l which imply N X cos{a(wj − wj)} w¨ = 2g2a3 , (2.111) j sin3{a(w − w )} k6=j j k in wi = xi and wi = zi.

Chapter 2 26 Section 2.6 A. R. Philip Soliton solutions to Calogero-Moser systems

2.7 Hydrodynamics and Solitons

Hydrodynamic wave motion is commonly described in terms of integrable nonlinear partial diﬀerential equations, P[ u(x, t) ] = 0, such as the Korteweg-de Vries (KdV) equation

ut − 12cuux + cuxxx = 0, (2.112)

the Burger’s Hopf (BH) equation

ut = 2cuux + cuxx, (2.113) or the Benjamin Ono (BO) equation

ut + uux + Huxx = 0, (2.114) where H denotes the Hilbert transform. Akin to the many-body Hamiltonian systems considered previously, these integrable systems retain Lax representations and Bäcklund transformations; and their integrability signifies that they possess an infinite number of conservation laws/conserved quantities. The latter suggests that they could in some sense be construed as finite integrable many-body systems in the limit N → ∞, with N denoting the number of bodies. Indeed the KdV equation, for instance, may in this limit be recovered from the finite Kac-van Moerbeke system of equations

x˙ n = exp{xn+1} − exp{xn−1} n = 1, ..., N − 1, (2.115)

which can be embedded into an integrable many-body Hamiltonian system that is a modiﬁed Toda lattice, as established by J. Moser [26]. Remark. The Toda lattice itself provides another embedding, but only in the limit N → ∞.

On solitons in hydrodynamic systems One should be aware that there is no universal agreement as to the definition of the soliton, or precisely what properties the term should include, whence varying conventions are present in the literature. However, the term and concept of the soliton derives from the solitary wave originally described by J. Scott Russell in 1844, which pertains to a phenomenon particular to nonlinear fluids, wherein the competing effects of nonlinearity and dissipation conspire to beget a single localized wave of permanent form, traveling at constant speed. The term permanent here signifies more than just non-dissipating, as the wave-form remains unchanged even after collisions between solitary waves. The solitary wave acquired its first formal expression at the hands of Boussinesq in 1871 and Lord Rayleigh in 1876, and would later be rediscovered by Korteweg and de Vries in the form: 1 1√ u(x, t) = v sech2 v(x − vt − x ) , (2.116) 2 2 0 as a solution to their Korteweg de Vries equation. Observe that (2.116) constitutes a meromorphic function with poles located at [25] iπ 1 p (t) = x − vt − √ n + , (2.117) n 0 v 2

Chapter 2 27 Section 2.7 A. R. Philip Soliton solutions to Calogero-Moser systems

and that its evolution in time is determined by the motion of these poles. Further motivated by the fact that interaction leaves the form of each involved solitary wave intact, Kruskal in 1974 conjectured that the poles of solutions with many solitary waves should form a many-body dynamical system of interacting particles, and thence coined the term soliton. Indeed, in the independent work of H. Airault, H.P. McKean & J. Moser [9] and G.V. Choodnovsky & D.V. Choodnovsky [3], in and around 1977, it was established that the motion of the poles of (2.116) is governed by a Hamiltonian system of one particle on the complex plane; and more generally that the poles xi to the N-soliton solution

N X 2 u(x, t) ∝ sech k(x − xi) , (2.118) i belong to a N-body Hamiltonian system (that is periodic along the imaginary axis). Some authors seemingly retain the term soliton to apply only to these hyperbolic solutions. However similar results were in fact also derived for other types of meromorphic solutions to the KdV equation, such as the rational solution

N X 1 2 u(x, t) ∝ , (2.119) x − x i i and D.V. Choodnovsky obtained analog results even for other hydrodynamic equations [4], [5], [6], [7], [8], such as the Burgers Hopf and Benjamin Ono equations. In the following we adopt the view, that any meromorphic solution, whose evolution is determined by a many-body Hamiltonian system comprised of its poles, is a soliton solution.

2.7.1 Calogero-Moser solitons in hydrodynamic systems As determined by D.V. Choodnovsky, the poles of rational meromorphic solutions to the Burger’s Hopf [7] and the Benjamin Ono equations [8], form rational Calogero-Moser systems. Here we are able to generalize these results to the CM systems of class II, III and I+. The generalization is based entirely on the general properties of the special Lax function.

’Pole ansatze’ to the Burger’s Hopf equation

Let α(x) be a special Lax function with asymptotic coeﬃcient c−1, such that α(x) → c−1/x as x → 0; and consider the (nominally modiﬁed) Burger’s Hopf (BH) equation

ut = 2cuux + c−1cuxx. (2.120)

Then the following holds true:

Theorem 2.7.1. To the BH equation a class of solutions is constituted by the meromorphic functions

N X u(x, t) = α(x − xi) − iωxi , (2.121) i=1

Chapter 2 28 Section 2.7 A. R. Philip Soliton solutions to Calogero-Moser systems

for arbitrary N, if and only if their poles, xj : j = 1, ..., N , satisfy

N X x˙ i = c α(xi − xj) + iωxi. (2.122) j=1

A proof is given in C.2. The equations of motion for the poles, by a time-derivative, imply Hamilton’s equations of motion for the isolated Calogero-Moser system with Lax function α(x), as established in the proof of the lemma in appendix B.1. Thus the BH equation permits at least four types of soliton solutions, governed respectively by the Calogero Moser systems of class I, II, III and I+.

’Pole ansatze’ to the Benjamin Ono equation

Now let α(x) be a special Lax function with asymptotic coeﬃcient c−1 = i, and consider the (somewhat generalized) Benjamin Ono (BO) equation

ut + 2cuux + cH[uxx] = 0, (2.123) where H[f] denotes the Hilbert transform, see appendix C.1. Note that the following has been proven only for special Lax functions to system of class I, III and I+, as we’re not aware of a Hilbert transform for the hyperbolic Lax function α(x) = coth x. It holds that

Theorem 2.7.2. The meromorphic functions

N M X X u(x, t) = α(x − xi) − α(x − zn) − iωx, (2.124) i n for arbitrary N and M, form a class of solutions to the BO equation, if and only if their poles, xi : j = 1, ..., N and zn : n = 1, ..., M , satisfy the system of equations

 N M  X X  x˙ i = − 2c α(xi − xj) + 2c α(xi − zn) + iωxi,   j6=i n=1 (2.125) M N  X X  z˙ = 2c α(z − z ) − 2c α(z − x ) + iωz .  n n m n l n  m6=n l=1

A proof is given in C.3. The eqs. of motion we recognize as the Bäcklund transformations of the special CM systems. As proven in B.1 they imply that the two systems of poles reside in two separate isolated CM systems, of the type with special Lax function α(x).

Chapter 2 29 Section 2.7 Chapter 3

Soliton solutions to Calogero-Moser systems

In this chapter we make use of the analytic results, gathered in the previous chapter, for the derivation and analysis of soliton solutions to the classical AN−1 N-body Calogero- Moser systems (introduced in §2.2). Firstly, in §3.1 below, we derive a general soliton equation, applicable to systems of class I, II, III (rational, hyperbolic and trigonometric) and I+ (rational with external harmonic potential), from which a large number of soliton solutions to these systems can be immediately generated. Note that the derivation of the soliton equation is uniﬁed for these systems, as the equation exists by virtue of the general properties of the special Lax function (introduced in §2.4.2), as follows from the proof in appendix B.1. Remark. For above-mentioned CM systems the interparticle potential may be written, V (x) = −α2(x), in terms of an odd special Lax function, α(x), satisfying the special functional equation

α(x)α0(y) − α(y)α0(x) = α(x + y) α0(y) − α0(x) .

These facts guarantee the existence of the general form of the Bäcklund transformation in §2.6.2 (or eqs. (3.2a), (3.2b) below) between a pair of CM systems of the same class (as proven in appendix B.1). In the following it will be shown that this Bäcklund transformation under certain restrictions produces the soliton equation. Any solution to the soliton equation corresponds to a configuration of the N particle- coordinates, at some fixed time t, that is compatible with an M-soliton wave-form, where M < N. The particle momenta, that define the evolution of the wave-form configuration, is found from a complementary equation. The instantaneous configuration may be propagated into advanced times, for example, by use of the OP projection method (reviewed in §2.5). After which it is possible to plot a worldline diagram of the soliton solution. In the following we will examine numerous worldline diagrams corresponding to representative and instructive soliton solutions, which are found to possess two fundamental characteristics:

(i) they correspond to electrostatic configurations of M ’positive’ charges zn(t) ∈ C in orbit around the principal ’negatively’ charged N-particle system {xi(t) ∈ M ⊂ C}, where M is a 1-dimensional manifold embedded in C, such that the latter system is effectively reduced into M degrees of freedom. (ii) instantaneously they correspond only to configurations that are extrema of an electrostatic energy-function, (3.11) in §3.1.3, that does not account for interactions

30 A. R. Philip Soliton solutions to Calogero-Moser systems

between the zn (which hence need not be in an extremal conﬁguration). Note that extrema may include but are not limited to minimizing conﬁgurations.

Within the principal particle system {xi(t)}, a soliton wave is induced by each trajectory of positive charge zn(t). The solitons are found to be localized, but with regularly varying amplitude, and appears quite generally to travel with constant speed, even when the zn does not (ostensibly because the zn(t) trajectories imply that the amplitude of the wave diminishes as the speed of zn increases, and vice-versa). The ’superposition’ of single soliton solutions is neither linear nor trivial, given that the zn are mutually repulsive, however we have diﬃculty in verifying any phase shift of the soliton waves as they pass each other by. To investigate this properly it might be necessary plot the time evolution of the relative density ρ(x) of the particles. Unfortunately, we have not done this in the following.

∗ ∗ ∗

Then, in §3.2, we extend our previous results by deriving a semi-hydrodynamic expression for the soliton equation: an expression that is partly formulated in terms of hydrodynamic fields of density and flow-velocity. This allows the hydrodynamic limit (N → ∞) of the CM systems to be performed. In the limit the trajectories of zn become ’parallel’ to the principal system (parallel to the real line), from which it follows that the corresponding soliton waves travel at constant speed with constant amplitude. Remark. The superposition of single-soliton wave-forms, and in particular the phase shift, should be clarified by the fact that solutions to the Benjamin Ono equation explicitly satisfy non-linear superposition. + Note that the derivation is unified for the AN−1 CM systems of class I, III and I , as it is made possible by the general properties of the special Lax function. Remark. The properties imply that the dynamics of certain soliton solutions to the Benjamin Ono equation is governed by CM systems of type I, III or I+. These systems would be composed of the poles of the soliton solutions, and they explicitly satisfy the Bäcklund transformation mentioned in the previous remark (see §2.7).

∗ ∗ ∗

In conclusion of the chapter, in §3.3.2, we first summarize our general method of §§3.1, 3.2, and in discuss the prospects of a soliton equation for the more general elliptic class IV CM system. The class IV system itself is not covered by the method, but in light of the summary we provide a few suggestions as to how it might in the future be included, and briefly discuss related matters in need of clarification.

Chapter 3 31 Section 3.0 A. R. Philip Soliton solutions to Calogero-Moser systems

3.1 Solitons from Bäcklund transformations of CM systems

As discussed earlier in §2.7, a soliton solution to a hydrodynamic system is meromorphic, and its evolution is governed by a many-body Hamiltonian system, comprised of its poles. In particular, an M-soliton solution corresponds to a Hamiltonian system of M particles on the complex plane. We also mentioned therein, that integrable hydrodynamic systems can be conceived of as integrable N-body systems in the limit N → ∞. In turn, the soliton solutions can be considered to connect a pair of many-body Hamiltonian systems, and given that solitons are permanent, to effectively reduce the infinite degrees of freedom (d.o.f.) present in the fluid, into the finite number of d.o.f. in the pole system. Thereby it seems reasonable to conjecture that, for an already finite many-body system of N particles, the M-soliton solutions should be governed by a many-body Hamil- tonian system of M particles on the complex plane, where M < N. To generate the soliton solutions, we would then need equations that connect the soliton solutions of the N-particle system, with solution-configurations of the M-particle system. This would seem to warrant a Bäcklund transformation.

∗ ∗ ∗

In the following we use the Bäcklund transformations, of §2.6.2, to derive soliton solutions to the isolated Calogero-Moser systems of class I, II III and I+ in a uniﬁed manner. First we describe the abstract general procedure applicable to all of the classes, and then provide the concrete formulas particular to each class.

3.1.1 General procedure

For convenience in the following we write α˜ = iα, where α˜ is the Lax function

Consider a pair of particle systems on the complex plane, {xj ∈ C : j = 1, ..., N} and {zn ∈ C : n = 1, ..., M}, that are both governed by the CM Hamiltonian

1 X X0 H = p2 + ω2q + V (q − q ),V (x) ∈ {I, II, III}, (3.1a) 2 i i i k i jk

with qi = xi and qi = zn. Let α˜(x) = iα(x) be the special Lax function that satisﬁes V (x) =α ˜(x)˜α(−x) = −α(x)α(−x) = α2(x). Then a subset of the solutions to each system, can be connected by a Bäcklund transformation on the form

N M X X x˙ j = − i α(xj − xk) + i α(xj − zn) + iωxj, (3.2a) k6=j n M N X X z˙n = i α(zn − zm) − i α(zn − xl) + iωzn. (3.2b) m6=n l

Remark. Note that ω = 0 for all but the class I interparticle potential V (x) = 1/x2.

Chapter 3 32 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

For convenience call { xi } the principal system, and { zn } the auxiliary system. Setting M < N, the auxiliary system, via the Bäcklund transformation, reduces the effective degrees of freedom in the principal system. Then the principal solutions, xi(t), allowed by the Bäcklund transformation, cannot be arbitrary and independent, but must be finely adjusted by the fewer particles and degrees of freedom in the auxiliary system. Moreover, if the principal system is constrained to the real line (and is non-divergent), the xi(t) correspond to permanent coordinated wave motion of the particles in the principal system. Constraining the system to the real line amounts to selecting an initial configuration of xi(0), x˙ i(0) ∈ R, ∀i, after which the Bäcklund transformation prevents each auxiliary zn and z˙n from being real simultaneously (if zn is real, z˙n must be complex). For xi, x˙ i real, (3.2a) may be separated into its real and imaginary parts

N M X X α(xj − xk) = Re α(xj − zn) + ωxj, (3.3a) k6=j n=1 M X x˙ j = − Im α(xj − zn), (3.3b) n=1 where (3.3a) constitutes our soliton equation. Again, note that ω = 0 unless α(x) ∝ 1/x. Given initial {zn(0)}, (3.3a) determines {xi(0)} corresponding to a M-soliton solution of the principal system. The {xi(0)} inserted into (3.3b) in turn determine the {x˙ i(0)}, and the auxiliary momenta {z˙n(0)} are determined by {xi(0)} inserted into (3.2b) (which demonstrates that not all solutions to the auxiliary system are permitted, but only a certain tuned subset of z, z˙, even if M = 1). Analytic solutions to (3.3a) are not known, except in the case that M = 0 (see [1]). But having solved it (approximately) at one instant in time, given that the CM systems are integrable, the evolution of the soliton solutions xi(t) and zn(t) are independently determined by the Olshanetsky-Perelomov projection method (reviewed in §2.5).

3.1.2 I+: Rational model on the line Although the isolated rational (class I) CM system admits soliton solutions by the pre- scribed procedure, these constitute a matter of mere academic interest, as the particles in the model drift apart, or disperse, with time. This would appear rather ill-suited for the study of permanent wave phenomena. Instead we consider the class I+ system, i.e. the rational CM system with a harmonic external potential, operating on the real line conﬁguration space:

N N 1 X X0 1 2 H = x˙ 2 + ω2x2 + g2 , x ∈ R. (3.4) rCM 2 i i x − x i i=1 i,j=1 i j For N > M, the Bäcklund transformation

N M X 1 X 1 x˙ i = − ig + ig + iωxi, (3.5a) xi − xj xi − zn j6=i n=1 M N X 1 X 1 z˙n = ig − ig + iωzn, (3.5b) zn − zm zn − xk m6=n k=1

Chapter 3 33 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

connects the real principal I+ system to an auxiliary I+ system in M complex coordinates zn ∈ C. Since xi and x˙ i are real, (3.5a) may be separated into its real and imaginary parts, to yield the soliton equations to our principal I+ model

N M X 1 X 1 ωxi = g − g Re , (3.6a) xi − xj xi − zn j6=i n=1 M X 1 x˙ = −g Im . (3.6b) i x − z n=1 i n

Provided an initial conﬁguration, {zn(0)}, the {xi(0)} are determined by (3.6a). After solving (3.6a) approximately, the {xj(t)} and {zn(t)} may separately and independently be obtained from the projection/OP procedure, i.e. from the Lax function pair choice α(x) = i/x, β(x) = −1/x2 they correspond to the eigenvalues of sin ωt ig sin ωt Xnm(t) = δnm wn(0) cos ωt +w ˙ n(0) + 1 − δnm , (3.7) ω ω wn(0) − wm(0)

for wj = xj and wj = zj respectively.

In passing, we would like to mention an interesting fact that we found. For simplicity we let ω = 0. Then it holds, of course, that

 N  2 X 1 3  x¨i = g ,  xi − xj  j6=i (3.8) M  2 X 1 3  z¨n = g .  zn − zm  m6=n

However, since it is possible to write

M M X xi − z¯n X Im zn x˙ i = Im 2 = 2 , (3.9) n |xi − zn| n |xi − zn|

an alternative expression for x¨i is given by

M X Imz ˙n 2 Im zn x¨i = 2 − 3 ∂t|xi − zn| . (3.10) n |xi − zn| |xi − zn| This expression shows that the acceleration of each particle depends on two parts, one essentially being the velocity of the imaginary part of zn, and the other the change in absolute distance between the particle xi and the zn. The latter is clearly of greater signiﬁcance when xi and zn are close, supposing that Im z ∼ Imz ˙.

Chapter 3 34 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

3.1.3 I+: Examples of soliton solutions

In the following we set g = ω = 1.

I+: 1-soliton solutions. In solving (3.6a) numerically1 for 1 auxiliary particle at z(0) = 1+i and 40 particles in the principal system, and separately propagating the solutions to the principal system and the complex auxiliary parameter, by the OP projection method, the worldline diagram of ﬁg. 3.1 can be assembled. Note in the diagram that the real x-axis has been rescaled by half the natural width of the particle system, R = p2gN/ω (see [1]). By inspection it appears that the soliton wave propagates along the real axis in coordination with the complex orbit z(t), where, for Im z(t) > 0 the soliton wave and z(t) travel leftwards, while for Im z(t) < 0 they travel rightwards. Although the speed of the auxiliary particle along the real axis is not constant, since z¨(t) = −ω2z(t), the soliton wave appears to propagate with constant speed.

0.8

π 0.6 t/2 ω 0.4

0.2

0 1.5 1 1.5 0.5 1 0 0.5 -0.5 0 iy -0.5 -1 -1 x/R -1.5 -1.5 Figure 3.1: World lines for real I+ system of N = 40 particles (in black, at iy = 0) with g = ω = 1, and the complex trajectory z(t) of the auxiliary particle (in blue), as determined by the 1-soliton solution with initial value z(0) = 1 + i (indicated by red dot). The real axis has been divided by R = p2gN/ω and the time axis by one period of the solution: 2π/ω. Observe that the trajectory of the auxiliary particle coordinates perfectly with the propagation of the soliton excitation through the real I+ system.

The trajectory of the auxiliary particle, z(t), completes an elliptical orbit on the complex plane. This can be seen more readily from fig. 3.2a, where z(t) has been drawn on the complex plane for all t ∈ 0, 2π (it’s essentially a bird’s eye view of fig. 3.1), and the extent of the real principal system is indicated by the black line on the real axis iy = 0. The elliptical orbit implies that the entire system, auxiliary parameter + real particles, is periodic in time. The period is generally given by T = 2π/ω [1]. In addition, in fig. 3.2b we show the real {xi(t)} plotted against t, i.e. the wordlines of just the real principal system as given by the above-specified 1-soliton solution. The soliton excitation corresponds to the leftward or rightward shifts in the worldlines, that propagate through the system. The shifts appear to be regularly spaced in time (until the extremes of the system), whence the soliton excitation appears to travel with constant speed within the system. Note the resemblance to a Newton’s cradle.

1For the task we used the function ’fsolve’ in MATLAB.

Chapter 3 35 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

1.5 1 0.9

1 0.8

0.7 0.5 0.6 π

iy 0 0.5 t/2 ω 0.4 -0.5 0.3

-1 0.2

0.1 -1.5 0 -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 x/R x/R (a) ﬁg 1 (b) ﬁg 2

Figure 3.2: Diagrams related to fig. 3.1, see fig. 3.1 for further details. a) The orbit z(t) of the auxiliary particle, drawn on the complex plane for all times t. Initial point indicated by red dot at z(0) = 1/R + i, and the extent of the principal I+ system is represented by the black line at iy = 0. Provides a ’birds-eye view’ of fig. 3.1. b) World lines of the principal I+ system on the real line, as determined by the 1-soliton solution.

Fig. 3.3 provides examples of 1-soliton solutions with initial z(0) closer to and, respectively, farther from the real axis. The former yields a tighter ellipse and more pronounced soliton wave, while the latter results in exactly the opposite. Thus the degree of separation between the auxiliary and principal particles is correlated with the amplitude of the soliton solution, which is suggestive of electrostatic attraction between particles of opposite charge. Indeed as pointed out by Abanov et al. in [1], the rational soliton equation corresponds to extremal conditions of the electrostatic energy-function

N 2 N N M X ωx X0 g X X 2 E = i − g ln |x − x | + ln |x − z | , (3.11) 2 i k 2 i n i ik i n where xi ∈ R. Our soliton solutions then correspond to extrema of E, and electrostatic attraction is facilitated by the 2-dimensional Coulomb potential V (xi−zn) = ln |xi − zn| = (1/2) ln(xi − zn)(xi − z¯n) . Take note that ∂V (x − z ) 1 1 1 1 i n = + = Re , (3.12) ∂xi 2 xi − zn xi − z¯n xi − zn whence it should hold that

N M ∂E X 1 X 1 = ωxi − g + g Re , (3.13) ∂xi xi − xk xi − zn k6=i n=1 which returns (3.6a) from the extremal condition ∂E/∂xi = 0. We conclude that the positively charged z(t) along its orbit attracts the system of negative charges on the line, and directly induces the soliton wave, by the imparting of a directed momentum. Though one should not imagine that all possible electrostatic conﬁgurations correspond to soliton solutions. Instead it is the case, that for every z(0) the soliton solution yields enough momentum z˙ to put the soliton parameter into a kind of free fall around the principal system.

Chapter 3 36 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

1.5 1 0.9

1 0.8

0.7 0.5 0.6 π

iy 0 0.5 t/2 ω 0.4 -0.5 0.3

-1 0.2

0.1 -1.5 0 -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 x/R x/R (a) ﬁg 1 (b) ﬁg 2

1.5 1 0.9

1 0.8

0.7 0.5 0.6 π

iy 0 0.5 t/2 ω 0.4 -0.5 0.3

-1 0.2

0.1 -1.5 0 -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 x/R x/R (c) ﬁg 3 (d) ﬁg 4

Figure 3.3: Diagrams related to 1-soliton solutions of I+ system with N = 40 and g = ω = 1, as determined by initial (a), (b) z(0) = 0.3(4/R + i) and (c), (d) z(0) = 1.4(2/R + i) respectively. (Left) Orbit of auxiliary particle, z(t), drawn on the complex plane for all times t. Red dot indicates z(0) and the black line shows the width of the I+ system. (Right) Wordlines of the I+ system on the real line. Observe that z(0) closer to the real system (c), (d) yields a tighter orbit and more pronounced wave, while z(0) farther away (a), (b) yields the opposite.

The present situation corresponds to electrostatics rather than electrodynamics, since all interaction is instantaneous. Electrodynamics would perhaps appear from similar treatment of soliton solutions to the relativistic generalization of the rational Calogero Moser system (see [11] for more on these systems).

A tightening orbit z(t) could also be the result of increasing the number of particles in the system, or of decreasing the value of ω. Thus, in the limit N → ∞, ω → 0, N/R = const., the two halves of the ellipse should disconnect and transform into straight lines, parallel to the real axis, with the left- and right moving solitons essentially becoming separate entities. In this limit the soliton excitations travel with constant velocity and constant amplitude, as z¨ = Im(z ˙) = 0. That the solitons seem to maintain constant speed more generally, we conjecture should be a result from the fact that larger Rez ˙ coincides with larger Im z, and consequently a weaker electrostatic impact upon the principal system (i.e. we conjecture that these eﬀects cancel out).

Chapter 3 37 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

I+: 2-soliton solutions and soliton interaction. To acquire some conception of soliton interaction, we should also examine the typical behaviour of 2-soliton solutions. Solving for 40 principal particles on the real line and the fairly representative initial conﬁguration z(0) = [z1(0), z2(0)] = [6 + 0.4i, 0.7 − i] of two auxiliary particles, and then propagating to advanced times using the OP projection method, results in the world line diagrams of ﬁg. 3.4:

0.8

π 0.6 t/2 ω 0.4

0.2

0 1.5 1 1.5 0.5 1 0 0.5 -0.5 0 iy -0.5 -1 -1 x/R -1.5 -1.5 (a)

1.5 1 0.9

1 0.8

0.7 0.5 0.6 π

iy 0 0.5 t/2 ω 0.4 -0.5 0.3

-1 0.2

0.1 -1.5 0 -1 0 1 -1.5 -1 -0.5 0 0.5 1 1.5 x/R x/R (b) (c)

Figure 3.4: Diagrams related to 2-soliton solution of I+ system of 40 particles and g = ω = 1, as determined by initial zn(0), n = 1, 2, where z(0) = [ 6 + 0.4i, 0.7 − i ]. a) Worldlines of real I+ system (in black, at iy = 0) and complex orbits of the auxiliary particles, z1(t), z2(t) (in blue). b) Orbits zn(t) drawn on the complex plane for all times + t, with zn(0) indicated by red dots, and width of the I system on the real line indicated by black line. c) Worldlines of real I+ system on the real line.

Note first, that adding solitons does not alter the periodicity of the system, which rather is a consequence of the action of the harmonic external potential on the complex plane. The auxiliary parameters, zi(t), here trace two separate ellipses, which seem to correspond roughly to the trajectories of their single particle solutions. This is representative only of solutions where the zi(t) are sufficiently removed from each other at all times. More generally the trajectories will be deformed from their single particle analogs, either slightly or significantly, depending on the degree of interaction between the zi(t). How- ever, even when the zn are safely separated, their associated soliton waves must at some

Chapter 3 38 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

point collide. This soliton interaction is pictured in the interior of ﬁg. 3.4c). It is evident that the form and identity of each soliton wave is preserved after interaction.

The auxiliary trajectories are not always so well behaved. If for any time they are sufficiently close to each other, or collide, rather more peculiar orbits may appear. An example with such zn(t)-collision is shown in figs. 3.5a) and 3.5b). Though no longer elliptic, the trajectories remain periodic with the same period. Note also that if the zi are close enough to interact but are separated by the principal system, they appear to be shielded from each other, see figs. 3.5c), 3.5d). This might be explained electrostatically by the fact that each zn brings along with it a concentration of negative charge, which should combine between them, and act as isolation.

1.5 1

0.9

1 0.8

0.7 0.5 0.6 π 0.5 iy

0 t/2 ω 0.4 -0.5 0.3

0.2 -1 0.1

-1.5 0 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 x/R x/R (a) ﬁg 1 (b) ﬁg 2

1.5 1

0.9

1 0.8

0.7 0.5 0.6 π 0.5 iy

0 t/2 ω 0.4 -0.5 0.3

0.2 -1 0.1

-1.5 0 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 x/R x/R (c) ﬁg 3 (d) ﬁg 4

Figure 3.5: Diagrams related to 2-solitons solutions of real I+ system with g = ω = 1, as determined by initial (a), (b) z(0) = [ 0.9i, 0.7i ] and (c), (d) z(0) = [ 6 + 0.15i, 6 − 0.15i ] respectively. (Left) Complex orbits of auxiliary particle, z(t), drawn on the complex plane for all times t. Red dot indicates z(0) and the black line shows the width of the I+ system. (Right) Wordlines of the I+ system on the real line.

I+: N-soliton solutions. Solutions for more solitons follow the same rules as outlined above. The trajectories of the M number of auxiliary zn are such that they by electrostatic attraction eﬀectively

Chapter 3 39 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

reduce the N degrees of freedom in the principal system, into the M degrees of freedom in the auxiliary system. The shapes of the trajectories depend on the degree of interaction between the zn: those with little interaction follow essentially their expected single particle ellipse, while those that interact are deformed in some manner. Examples of both situations, at the same time, is provided by ﬁg. 3.6. Note especially, in (c) that two uppermost trajectories are inversely identical, and consequently that the uppermost trajectory is not elliptic (its continuation is not part of the lowest-going trajectory).

1.5 1

0.9 1 0.8

0.7 0.5 0.6 π

iy 0 0.5 t/2 ω 0.4 -0.5 0.3

-1 0.2 0.1

-1.5 0 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 x/R x/R (a) ﬁg 3 (b) ﬁg 4

1.5 1

1 0.9 0.8

0.5 0.7

0.6 π iy 0 0.5 t/2 ω 0.4 -0.5 0.3

-1 0.2 0.1

-1.5 0 -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 x/R x/R (c) ﬁg 3 (d) ﬁg 4

Figure 3.6: Diagrams related to 2-soliton solutions of real I+ system with g = ω = 1, as determined by initial (a), (b) z = [ 0.9i, 0.7i, 0.5(2−i)] and (c), (d) z = [ 0.9i, 0.7i, 1.4(2+ i)] respectively. (Left) Complex orbits of auxiliary particles, zn(t), drawn on the complex + plane for all times t. Red dots indicate zn(0). (Right) Wordlines of the I system on the real line. Note that the uppermost trajectory in (c) is not elliptic, instead, the two uppermost trajectories are inverse w.r.t. the real axis (so the 2:nd uppermost trajectory is also the lowest-going trajectory).

Chapter 3 40 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

3.1.4 II: Hyperbolic model Given that the hCM system is gravely repulsive, and we at present possess no type II Bäcklund transformation with a ’gathering’ external potential, we are currently unable to produce any meaningful solution plot for a hyperbolic CM model. However the theory remains valid, and we describe it below:

Consider the principal system

N 1 X X0 1 H = p2 + g2a2 , (3.14) hCM 2 j sinh2 {a(q − q )} j jk j k

in canonical coordinates qi, pi ∈ R, related to motion on a hyperbolic manifold by the canonical transformation aqi = (1/2) lnq î, pi =q îpî. The {qi} represent, essentially, the hyperbolic analog to angles of complex coordinates on the unit circle. Soliton solutions are found from the BT

N M X X x˙ j = − iga coth{a(xj − xk)} + iga coth{a(xj − zn)}, (3.15) k6=j n M N X X z˙n = iga coth{a(zn − zm)} − iga coth{a(zn − xl)}, (3.16) m6=n l where, since xi and x˙ i are real, (3.15) may be divided into the real and imaginary parts

N M X X ga coth{a(xj − xk)} = ga Re coth{a(xj − zn)}, (3.17) k6=j n M X x˙ n = −ga Im coth{a(zn − xl)}. (3.18) l

Note that the zn ∈ C, so they are not simply transformed to a hyperbolic manifold by the analog canonical transformation. The soliton equation (3.17) again must be solved approximately at t = 0, whereafter the solutions at arbitrary time, on the hyperbolic manifold, correspond to the eigenvalues of

X(t) = exp{aL(0)t} exp{2aQ(0)} exp{aL(0)t}, (3.19) where   Qnm = δnmqn(t), (3.20) iga  Lnm = δnmpn(t) + 1 − δnm , sinh {a(qn − qm)} as per the OP projection method.

Chapter 3 41 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

3.1.5 III: Trigonometric model on the unit circle Provided the principal system

N 1 X X0 1 H = x˙ 2 + g2a2 , (3.21) tCM 2 i sin2 {a(x − x )} i=1 ij i j

of particles on the unit circle, with each xi an angle (or real phase of a complex number), its soliton solutions are provided by the BT

N M X X x˙ i = − iga cot{a(xi − xj)} + iga cot{a(xi − zn)}, (3.22) j6=i n M N X X z˙n = iga cot{a(zn − zm)} − iga cot{a(zn − xk)}, (3.23) m6=n k

connecting it to an identical system, but of M < N complex ’angles’ zn. To translate the angles xi, to complex coordinates on the unit circle, we write xˆi = exp{2iaxi}. The analog transformation, zˆn = exp{2iazn} = exp{−2a Im zn} exp{2ia Re zn}, maps the zn to complex numbers of amplitude exp{−2a Im zn}. Since the xi and x˙ i are real, (3.22) may be separated into its real and imaginary parts

N M X X ga cot{a(xj − xk)} = ga Re cot{a(xj − zn)}, (3.24) k6=j n M X x˙ n = −ga Im cot{a(zn − xl)}. (3.25) l Yet again the soliton equation (3.24) must be solved approximately at t = 0, after which the solutions may be propagated onwards in time by way of the OP projection method. The complex coordinates xˆi(t) = exp{2iaqi(t)} of the particles on the unit circle, are then provided as the eigenvalues of

X(t) = exp{iaL(0)t} exp{2iaQ(0)} exp{iaL(0)t}, (3.26)

where   Qnm = δnmqn(t), (3.27) iga  Lnm = δnmpn(t) + 1 − δnm . sin {a(qn − qm)} Note that the trigonometric model on the unit circle, phrased in terms of complex ’angles’, essentially corresponds to an inﬁnite model on the line, for which it holds that Imz ˙n = 0 and Im zn(t) = const.. Hence each complex angle-trajectory, zn(t), maps to zˆn(t) = exp{2iazn(t)} = exp{−2a Im zn(t)} exp{2ia Re zn(t)}, which is a complex circle of radius exp{−2a Im zn(t)}.

Chapter 3 42 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

3.1.6 III: Examples of soliton solutions

In the following we set g = ω = a = 1.

III: 1-soliton solutions Solving (3.24) with N = 2, M = 1 for z(0) = 0.8(1−i) and, respectively, z(0) = 0.8(1+i), the worldline diagrams of ﬁg. 3.7 are obtained. The LHS of 3.7 shows the wordlines of the principal and auxiliary particles on the complex plane (the xˆi(t) and zˆ(t) against t), and the RHS shows the real angles of just the principal particles (the xi(t) against t). We have, for both respective solutions, that the trajectory of the auxiliary particle zˆ(t) traces a perfect circle, and that the soliton wave travels oppositely to the auxiliary particle: outside of the unit circle, zˆ(t) turns clockwise and the soliton wave turns anti- clockwise; and vice-versa for zˆ(t) on the inside. The angular velocity of the soliton wave corresponds to the velocity of z˙, which is constant since z¨ = 0. The time has here been chosen arbitrarily, so the continuation of the diagrams, though similar, is not periodic.

4.5 5 4 4 3.5

t 3 3

2 t 2.5 1 2 1.5 0 4 1 2 4 0.5 0 2 0 -2 0 Im -2 -4 -3 -2 -1 0 1 2 3 4 -4 -4 Re phase (a) ﬁg 1 (b) ﬁg 2

4.5 5 4 4 3.5

t 3 3

2 t 2.5 1 2 1.5 0 1 1

1 0.5 0 0 0 Im -4 -3 -2 -1 0 1 2 3 4 -1 -1 Re phase (c) ﬁg 3 (d) ﬁg 4

Figure 3.7: Worldline diagrams for 1-soliton solution of principal III system of 2 particles on the unit circle and g = ω = 1, as determined by initial (a), (b) z(0) = 0.8(1 − i) and (c), (d) z(0) = 0.8(1 + 1i). (Left) Worldlines for III system on the complex unit circle (in red) and trajectory zˆ(t) = exp{iz(t)} of the auxiliary particle (in blue). (Right) Real angles of III system on the unit circle. Note 2π-periodicity along the horizontal axis.

Chapter 3 43 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

Solutions with a greater number of particles in the principal system, such as the two examples of ﬁg 3.8, shows that the number of turns of zˆ(t) around the principal system, in a given time, increases with the number of particles. In the electrostatic interpretation (see §3.1.3), it is clear that the soliton solution corresponds to an initial z˙(0) such that the z-particle enters into ’free fall’ around the system of negative charge (if inside the unit circle, it falls outwards). Increasing negative charges on the unit circle, z˙ must increase as well, resulting in additional turns around the system.

4.5 5 4 4 3.5

t 3 3

2 t 2.5 1 2 1.5 0 4 1 2 4 0.5 0 2 0 -2 0 Im -2 -4 -3 -2 -1 0 1 2 3 4 -4 -4 Re phase (a) ﬁg 1 (b) ﬁg 2

4.5 5 4 4 3.5

t 3 3

2 t 2.5 1 2 1.5 0 4 1 2 4 0.5 0 2 0 -2 0 Im -2 -4 -3 -2 -1 0 1 2 3 4 -4 -4 Re phase (c) ﬁg 3 (d) ﬁg 4

Figure 3.8: Worldline diagrams for 1-soliton solution of principal III system as determined by initial z(0) = 0.8(1 − i), with g = ω = a = 1, of (a), (b) 3 and (c), (d) 5 particles on the unit circle, respectively. (Left) Worldlines for III system on the complex unit circle (in red) and trajectory zˆ(t) = exp{iz(t)} of auxiliary particle (in blue). (Right) Real angles of III system on the unit circle. Note 2π-periodicity along the horizontal axis.

Chapter 3 44 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

III: 2-soliton solutions and soliton interaction. From the 2-soliton solution with N = 3, z(0) = 0.6(1 + i), 0.6(1 − i), the diagrams of ﬁg. 3.9 are found. The circular trajectories in this particular case superpose perfectly, which remains true also when more particles are added to the principal system. It might look like the interior screw has changed handedness, but this is a trick of perspective: it does travel counter-clockwise on a circular orbit, as it should.

4.5 5 4 4 3.5

t 3 3

2 t 2.5 1 2 1.5 0 4 1 2 4 0.5 0 2 0 -2 0 Im -2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -4 -4 Re phase (a) ﬁg 1 (b) ﬁg 2

Figure 3.9: Worldlines for 2-soliton solution of principal III system of 3 particles on the unit circle, with g = ω = 1, as determined by initial z(0) = 0.6(1 + i), 0.6(1 − i). a) Wordlines of III system (in red) and trajectories zˆn(t) = exp{izn(t)} of soliton particles (in blue), on the complex plane. b) Worldlines of the real angles of principal III system on the unit circle.

However, this stable superposition of circular orbits only appears when the initial values of the auxiliary parameters are conjugate. If instead the real parts are the same but the imaginary parts not, the outer trajectory tends to ’wobble’, as in ﬁg. 3.10.

1.5 3 1

0.5 2 t 0 Im 1 -0.5

0 -1 2 2 -1.5 0 1 0 -2 Im -1 -2 -1 0 1 2 -2 -2 Re Re (a) (b)

Figure 3.10: Worldlines for 2-soliton solution of principal III system of 3 particles on the unit circle, with g = ω = 1, as determined by z(0) = [0.6(1 − i), 0.6(4 + i)]. a) Worldlines for principal III system (in red) and trajectories of auxiliary particles (in blue). b) Wordlines of (a) projected down onto the complex plane.

Chapter 3 45 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

In contrast, if the imaginary parts are the same but the real parts differ, we find severely unstable trajectories such as those of fig. 3.11. In figs. 3.11a and 3.11b the trajectory of one zn seem to open a gap in the principal system that the other can slip in through, and vice-versa. Note that the soliton wave doesn’t differ much from the conjugate case earlier. It is rather amazing that such apparent chaos of the zn trajectories should yield a well behaved soliton solution.

4.5 5 4 4 3.5

t 3 3

2 t 2.5 1 2 1.5 0 2 1

2 0.5 0 0 0 Im -4 -3 -2 -1 0 1 2 3 4 -2 -2 Re phase (a) ﬁg 1 (b) ﬁg 2

4.5

4 3.5 3 t

2 t 2.5 2

1.5 0 2 1 2 0 0.5 0 Im 0 -2 -2 Re -4 -3 -2 -1 0 1 2 3 4 phase (c) ﬁg 3 (d) ﬁg 4

Figure 3.11: Worldlines for 2-soliton solutions of principal III system with g = ω = a = 1, as determined by initial z(0) = [0.6(1−i), 0.6(4+i)], in (a), (b) 3 and, respectively (c), (d) 8 particles on the unit circle. (Left) Worldlines for III system on the complex unit circle (in red) and trajectories zˆn(t) = exp{izn(t)} of auxiliary particles (in blue). (Right) Real angles of principal III system on the unit circle. Note 2π-periodicity along the horizontal axis.

With more particles in the principal system, as shown in ﬁgs 3.11c and 3.11d, the auxiliary trajectories seem to stabilize, although they steadily decline towards the principal system (the inner one declines outwards), forming opposite funnel shapes. However, in the continuation of the diagram (not shown), when the zn are suﬃciently close to the principal system, a pair of gaps open and the zn pass through, but otherwise continue essentially on their earlier trajectories. Increasing the number of particles has thus only postponed the crossing of the unit circle. But this implies that in the limit N → ∞ the funnel-shapes should straighten out to cylinders of perfectly stable circular orbits (which

Chapter 3 46 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems for the rational model on the real line corresponds to a to a condition of suﬃcient density).

As a ﬁnal example, for a pair of zn(0) on the same side of the unit circle, the trajectories zˆn(t) ’wobble’ between orbits of varying radii.

4.5 5 4 4 3.5

t 3 3

2 t 2.5 1 2 1.5 0 4 1 2 4 0.5 0 2 0 -2 0 Im -2 -4 -3 -2 -1 0 1 2 3 4 -4 -4 Re phase (a) ﬁg 1 (b) ﬁg 2

Figure 3.12: Worldlines for principal III system in 3 particles on the unit circle, as determined by initial z(0) = 0.6(1−i), 1.2(4−i). a) Worldlines for principal III system (in red) and trajectories of auxiliary particles (in blue). b) Worldlines of the real angles of principal III system on the unit circle.

Chapter 3 47 Section 3.1 A. R. Philip Soliton solutions to Calogero-Moser systems

3.2 Solitons from semi-hydrodynamic Bäcklund transformations

In the following we strive to extend our results for the finite Calogero-Moser systems, for application to their infinite limit N → ∞. Then the principal system becomes tantamount to a liquid, described by fields of density and flow-velocity, while the auxiliary system remains a finite many-body system (these terms are introduced in §3.1). To this end we evidently require some formulation of the soliton equations, or rather the Bäcklund transformations, that connect classical dynamical fields with coordinates and momenta of a many-body system of particles. Such semi-hydrodynamic soliton equations have previously been derived for the rational and the trigonometric CM systems, by Abanov et al. in [1] and, respectively [2]. Their method is based on the relationship between the Benjamin Ono equation and Bäck- lund transformations between CM systems, as discussed here in §2.7. In the following we follow essentially their example, but take a more general approach that unifies the procedure for the CM systems of classes I, III and I+. Remark. One should note that the Benjamin Ono equation might not be the only possible way to to derive the semi-hydrodynamic equations, in the manner that will be explained in the following, as the CM Bäcklund transformations are related in a very similar manner to solutions to the Schrödinger equation. Hence it might well be possible to develop an analog derivation based on the Schrödinger equation in place of the Benjamin Ono equation, which could include the hyperbolic class II system, and possibly serve to clarify the relation between classical- and quantum mechanical CM systems.

3.2.1 Bidirectional Benjamin Ono equation From §2.7 we are aware that the Bäcklund transformations (BTs) between CM systems can be recovered from solutions to the Benjamin Ono (BO) equation, in the taking of residues z → xi, z → zn, where {xi} and {zn} are two distinct set of poles to the solution. This connection between the BTs and the Benjamin Ono equation is independent of N and M, and hence should remain true in the limit N → ∞ with M kept ﬁnite. However, there are reasons not to use the BO equation directly. Indeed, our proofs in appendix C.3 technically require the two sets of poles to be restricted to respective sides of the real axis, whereas we would like that xi ∈ R. Hence we proceed by avoiding the Hilbert transform altogether, and instead consider the solutions

N M − X + X u = i α(x − xi) − iωx, u = −i α(x − zn), (3.28) i=1 n=1 to the bidirectional Benjamin Ono (2BO) equation:

1 1 1 u = uu − igu+ + igu− = ∂ u2 + ig∂ u− − u+ , (3.29) t x 2 xx 2 xx 2 x x x x where u = u− + u+. This allows us to withhold assumptions and restriction on the sets of poles {xi}, {zn}, while the proof in appendix C.3 remains both valid and applicable to the present case, with minimal modiﬁcation.

Chapter 3 48 Section 3.2 A. R. Philip Soliton solutions to Calogero-Moser systems

3.2.2 General procedure PN − First we introduce the density ﬁeld ρ(x) = i δ(x − xi). In terms of ρ(x), u (z) may be rewritten as Z ∞ u−(z) = ig dx ρ(y)α(z − x) − iωz, (3.30) −∞ for z that are not on the real line, z 6= x ∈ R (note that the integration limits should be 0 → 2π for the trigonometric class III system). On the real line (3.30) it is discontinuous, such that u−(x ± i0) = ±πgρ(x) − i(πgρH (x) + ωx), (3.31) where ρH is the Hilbert transform of ρ, see appendix C.1. (For systems of class I and I+ it is written p.v. Z ∞ ρH (x) = f(z)α(x − z)dz, (3.32) π −∞ while for system III we have that p.v. Z 2π ρH (x) = f(z)α(x − z)dz. (3.33) π 0 We are not aware of an analog Hilbert transform for the hyperbolic II system, and there is reason to suspect that having α(x) = coth(x) above would lead to a divergent integral.) Thereby u−(x + i0) − u−(x − i0) = 2πgρ(x), (3.34) u−(x + i0) + u−(x − i0) = −2i(πgρH + ωx). (3.35) In addition, since the function u+(x) is continuous on R, it holds that u(x + i0) − u(x − i0) = 2πgρ(x). (3.36) The appropriate generalization to taking the residue of the BO equation, for when the poles xi have been replaced by the ﬁeld ρ(x), is to compare its values above and below the real line. Thus, if we introduce the notation ∆f = f(x + i0) − f(x − i0) for an arbitrary function f, we should consider 1 1 ∆u ˙ = ∂ ∆u2 + ig∂ ∆u− − ∆u+ = ∂ 2u+∆u− + ∆(u−)2 + ig∂ ∆u− . (3.37) 2 x x x x 2 x x x

Using (3.36), whereby ∆u ˙ = 2πρt, (3.34) and (3.35), we ﬁnd that + − − 2πρt = πg ∂x 2ρu + ρ u (x + i0) + u (x − i0) + ig ∂xρx + H √ = πg ∂x ρ 2u − 2iπgρ − 2iωx + 2ig∂x log ρ . (3.38)

The purpose of writing the expression in this manner, is that the continuity equation

ρt + ∂x(ρv) = 0, (3.39)

+ H √ is recovered if the ﬂuid velocity is v = 2πg u − iπgρ − iωx + ig∂x log ρ .

Chapter 3 49 Section 3.2 A. R. Philip Soliton solutions to Calogero-Moser systems

Remark. Note that v denotes the rate of change of ρ(x), not the velocity of the soliton wave solution. Then it follows, from the expression for v(x), that

1 √ X u+(x) = v + iπgρH + iωx − ig∂ log ρ = −ig α(x − z ), (3.40) 2πg x n n which constitutes our semi-hydrodynamic extension of the Calogero-Moser Bäcklund transformation. It connects ﬁeld solutions ρ(x), v(x) of the principal system, to many- body solutions zn(t), z˙n(t) to the auxiliary many-body system. Assuming ρ(x) and v(x) to be real valued functions, we can separate the semi- hydrodynamic Bäcklund transformation into its real and imaginary parts

M H √ X g πρ − ∂x log ρ − ωx = −g Re α(x − zn), (3.41a) n=1 M X v = 2πg Im α(x − zn), (3.41b) n=1 which constitutes the semi-hydrodynamic generalizations of the soliton equations. For ﬁnite N these are in fact equivalent to the many-body soliton equations (3.3a), (3.3b) [1]. Remark. Be aware that our expression (3.41b) diﬀers from its equivalent in [1] by a factor of 2. I do not know why. There might be some error of calculation above. Remark. Note that ω = 0 for all but the I+ system, with α(x) ∝ 1/x.

The momenta z˙n are in turn given by

N X Z ∞ z˙n = i α(zn − zm) − i dxρ(x)α(zn − x) + iωzn, (3.42) m6=n −∞ which can be deduced from (3.2b).

3.2.3 I: Rational model The semi-hydrodynamic soliton equations for the rational Calogero-Moser model are as such:

M √ X 1 gπρH − ∂ log ρ − ωx = −g Re , (3.43) x x − z n n M X 1 v = 2πg Im . (3.44) x − z n n Note that, as the principal system holds an inﬁnite number of particles, that serve to constrain each other, it is not necessary to add an external potential to the model, as we had in §3.1.2. Hence we may set ω = 0. From the analysis of the ﬁnite class I model in (3.1.3), and in particular its extrap- olation to N → ∞, we know that the soliton parameters zn travel along straight lines with Imz ˙ = 0 (apart from those soliton solutions where the zn become close). Does this

Chapter 3 50 Section 3.2 A. R. Philip Soliton solutions to Calogero-Moser systems

criterion have some signiﬁcance? One might note that the ﬂuid velocity may be rewritten as

M M X x − z¯n X Im zn v = 2πg Im 2 = 2πg 2 (3.45) n |x − zn| n |x − zn| Then the change in ﬂuid velocity appears in two parts:

M X Imz ˙n 2 Im zn v˙ = 2πg 2 − 3 ∂t|x − zn| . (3.46) n |x − zn| |x − zn| Hence, for the well-behaved hydrodynamic soliton solutions, the first term vanishes, and the fluid velocity v(x) depends only on the change in distance between the points zn and x. If for each zn we integrate the second term over x symmetrically around zn, the terms should cancel to yield 0, i.e. ρ(|d − z|) decreases as quickly as ρ(|d + z|), for every d, which essentially is another way of saying that the wave form follows z perfectly, which is possible only if they both travel at constant speed. Were the z-particle instead to accelerate, the wave-length in front would be shorter than the wavelength behind, which is matched by the first term of (3.46) if Imz ˙n 6= 0, which has the same value for two points symmetrically around zn.

3.2.4 III: Trigonometric model The soliton solutions of the trigonometric model should be given by

M H √ X g πρ − ∂x log ρ − ωx = −ga Re cot{a(x − zn)}, (3.47) n M X v = 2πga Im cot{a(x − zn)}. (3.48) n

Chapter 3 51 Section 3.2 A. R. Philip Soliton solutions to Calogero-Moser systems

3.3 Concluding remarks on soliton solutions to the elliptic CM systems

3.3.1 Summary of the unified method In §3.1 we constructed a unified description of soliton solutions to the many-body CM systems of class I, II, III and I+. The construction is based on the common properties of their special Lax functions α(x). More specifically, since the interparticle potential of each system may be written V (x) = α(x)α(−x) = −α2(x), and since α(x) satisfies the special Lax function

α(x)α0(y) − α(y)α0(x) = α(x + y)α0(y) − α0(x), (3.49) it holds that the system of equations, for arbitrary N and M

N M X X x˙ i = − α(xi − xj) + α(xi − zn) + iωxi, (3.50a) j6=i n=1 M N X X z˙n = α(zn − zm) − α(zn − xj) + iωxi, (3.50b) m6=n j=1 forms a Bäcklund transformation between a pair of CM systems in N and M particles respectively, both with interparticle potential V (x). We prove this in appendix B.1. Remark. Note that ω = 0 for all but the class I+ system. Then, with M < N and the set of N particles restricted to the real line, the Bäcklund transformation implies the soliton equations

N M X X α˜(xi − xj) = Reα ˜(xi − zn) + ωxi, (3.51a) j6=i n=1 M X x˙ i = − Imα ˜(xi − zn), (3.51b) n=1 where α(x) = iα˜(x), whose solutions correspond to M-soliton solutions to an N-body CM system of class I, II, III or I+.

In §3.2 we extended this many-body description to also include soliton solutions to the hydrodynamic CM systems of class I, III and I+. The extension is also based on the properties of the special Lax function, α(x), which imply that the above-mentioned Bäcklund transformation appears from the residues of the (bidirectional) Benjamin Ono equation 1 1 1 u = uu − igu+ + igu− = ∂ u2 + ig∂ u− − u+ , (3.52) t x 2 xx 2 xx 2 x x x x where u = u− + u+, after insertion of solutions

N M − X + X u = i α˜(x − xi) − iωx, u = −i α˜(x − zn). (3.53) i n

Chapter 3 52 Section 3.3 A. R. Philip Soliton solutions to Calogero-Moser systems

This we have proven in appendix C.3. Therefrom we could derive a semi-hydrodynamic representation of the soliton equations

M H √ X g πρ − ∂x log ρ − ωx = −g Re α(x − zn), (3.54a) n=1 M X v = 2πg Im α(x − zn), (3.54b) n=1 applicable both for ﬁnite N and to the hydrodynamic limit of the CM systems, where N → ∞ with density constant.

3.3.2 Soliton solutions to elliptic CM model (IV) Now, according to Wojciechowski [22] the Bäcklund transformations

N M X X x˙ i = −ig ζ(xi − xk) + ig ζ(xi − zn), (3.55) k m M N X X z˙n = ig ζ(zn − zm) − ig ζ(zn − xk). (3.56) m k imply Newton’s eqs. of motion

X 0 X 0 x¨i = − ℘ (xi − xk), z¨n = − ℘ (zn − zm), (3.57) k m for a pair of free Calogero Moser systems of the general elliptic kind (type IV) iﬀ. N = M. A proof is given by A. Sciarappa in [31]. Therefrom, supposing xi(t) ∈ R for all times, the real and imaginary parts yield, as before

N M X X ζ(xj − xk) = Re ζ(xj − zn), (3.58a) k6=j n M X x˙ j = − Im ζ(xj − zn), (3.58b) n which could have constituted the soliton equations, generating soliton solutions, to the free elliptic CM system

N N 1 X X0 H = x˙ 2 + g2 ℘(x − x ). (3.59) CM 2 i i k i ik if M < N. However as far as we’re aware, this is not allowed, and there are a few more matters that remain unclear with respect to our earlier developments. Firstly one should note that, although the Weierstrass zeta function is deﬁned as ℘(z) = −ζ0(z), it does not exactly satisfy the special functional equation, but instead

ζ(x)ζ0(y) − ζ(y)ζ0(x) = ζ(x + y) ζ0(y) − ζ0(x) + Γ(x) − Γ(y), (3.60)

Chapter 3 53 Section 3.3 A. R. Philip Soliton solutions to Calogero-Moser systems

0 1 00 where Γ(x) = ζ(x)ζ (x)+ 2 ζ (x) [7]. Neither is ζ(z) a Lax function of the Calogero Moser systems, as V (x) = ℘(x) 6= ζ(x)ζ(−x). Though it can be found from ζ(z) = σ0(z)/σ(z), where σ is the Weierstrass sigma function, that

σ00(z) σ02(z) σ00(z) ζ0(z) = − = − ζ2(z) σ(z) σ2(z) σ(z) s s σ00(z) σ00(−z) = + ζ(z) + ζ(−z) , (3.61) σ(z) σ(−z)

wherefore ℘(z) =α ˆ(z)ˆα(−z), with s σ00(z) αˆ(z) = i ζ(z) + . (3.62) σ(z)

Before it can be concluded that αˆ(z) constitutes a CM Lax function, however, one must show that it satisﬁes the Lax functional equation

α(x)α0(y) − α(y)α0(x) = α(x + y) β(x) − β(y) (3.63)

1 00 for some β(x) = i ℘(x) + β0 = α (x)/α(x) + β0, with arbitrary constant β0.

3.3.3 Hydrodynamic soliton solutions to the elliptic model? To take the hydrodynamic limit of the type IV soliton equations above, we would like solutions to the Benjamin Ono equation for which its residues beget the elliptic Bäcklund transformations. This at present, we do not have. However, as found by Wojciechowski [22], the solutions

M Πn σ(z − zn) iε(λz−λ2t) ψ(z, t) = N e . (3.64) Πi σ(z − xi) PM inserted into the Schrödinger equation iψt = −ψxx + U(z)ψ with U(z) = n ℘(z − zn), yields residues that correspond to the elliptic Bäcklund transformations. Hence it might be possible to derive the hydrodynamic soliton equations from the Schrödinger equation instead.

3.3.4 Elliptic projection method and KKS construction? Finally, it would very much be of interest to have at hand a generalization of the OP projection method, applying to the elliptic CM systems. This would ﬁrst necessitate such a generalization of the KKS construction; and as the special CM systems derive from 0 geodesic motion in subspaces of the vector space XN = {X} of N × N traceless matrices X, with the rational CM systems constructed from free motion, and the hyperbolic systems from hyperbolic motion, it would be reasonable to conjecture that the elliptic 0 CM systems may be constructed from elliptic motion on XN . This problem has been discussed in a more general setting by E. Langmann in [32].

Chapter 54 Section .0 Appendix A

Appendix on Lax representations.

A.1 A formal construction of the Lax pair of an integrable system

Suppose that H is integrable, and that the integrals of motion H1, ..., HN are known a priori. Then the Lax representation of H can be formally constructed, by ﬁrst arranging the Hn into a diagonal matrix, ˆ L = diag Hn : n = 1, ··· N . (A.1)

Note that similarity transformations preserve eigenvalues. Were Lˆ to be put though a similarity transformation in some invertible matrix U(t)

L = ULUˆ −1, (A.2)

the eigenvalues λn of L would still be identical to the integrals of motion, λn = Hn, ˙ wherefore λn = 0. Thus, the time derivative of the eigenvalue equation Lun = λnun becomes ˙ Lun + Lu˙ n = λnu˙ n. (A.3)

Nothing more can be done, without supposing the eigenvectors un to evolve according to

u˙ n = −Mun, (A.4)

where M is some N × N matrix. Then it must be that ˙ L − [L, M] un = 0, (A.5)

and the Lax evolution equation is recovered. Since the elements of L are functions of qn and pn, and λn = Hn, the Lax evolution equation must imply Hamilton’s equations of motion for H.

55 A. R. Philip Soliton solutions to Calogero-Moser systems

A.2 Derivation of the CM Lax representations from Calogero’s ansatz

Calogero’s ansatz for the Calogero Moser Lax pair reads [11]   L = δ p + (1 − δ )α(q − q ),  nm nm n nm n m N (A.6) X  M = δ β(q − q ) + (1 − δ )γ(q − q ).  nm nm n l nm n m  l6=n

For present convenience we expand the Lax evolution equation L˙ = [L, M] into   X  L˙ = L M − L M ,  nn nl ln ln nl l6=n (A.7) X  L˙ = L − L M + L M − M + L M − L M .  nm nn mm nm nm mm nn nl lm lm nl  l6=m,n Then, insertion of Calogero’s ansatze (A.6) into (A.7) yields X for n = m: p˙n = α(qn − ql)γ(ql − qn) − α(ql − qn)γ(qn − ql) , (A.8) l6=n

0 for n 6= m: q˙n − q˙m α (qn − qm) = X X pn − pm γ(qn − qm) + α(qn − qm) β(qm − ql) − β(qn − ql) + l6=m l6=n X + α(qn − ql)γ(ql − qm) − α(ql − qm)γ(qn − ql) (A.9). l6=n,m A multiparticle Hamiltonian ought to have q˙ = ∂H = p . Taking this as a requirement, n ∂qn n (A.9) holds if the following conditions are satisﬁed ( γ(q) = α0(q), (A.10a) β(q) = β(−q), (A.10b) and α(qn − qm) β(qm − ql) + β(qn − ql)

= − α(qn − ql)γ(ql − qm) + α(ql − qm)γ(qn − ql). (A.10c) Condition (A.10a) inserted into (A.8) reproduce Newton’s equation

X 0 p˙n = − V (qn − qm), (A.11) m6=n with V (q) = V (−q) = α(q)α(−q). Thus (A.11) and q˙n = pn together form Hamilton’s equations of motion for the Hamiltonian 1 X X0 H = p2 + V (q − q ). (A.12) 2 n n m n nm

Chapter A 56 Section A.2 A. R. Philip Soliton solutions to Calogero-Moser systems

Finally, in writing x = qn − ql and y = ql − qm, the application of condition (A.10b) to condition (A.10c) results in the functional equation α(x)α0(y) − α(y)α0(x) = β(x) − β(y), (A.13) α(x + y) which may be regarded as a functional representation of the Lax pair associated with the Hamiltonian.

A.3 Local analysis of the functional equation and derivation of the general CM potential

Following Calogero (ch. 2.1.11 of [11]) we study the local structure of the functional equation

α(x)α0(y) − α(y)α0(x) = α(x + y)β(x) − β(y). (A.14)

Our purpose herewith is to derive certain facts about the Lax functions α(x) and β(x). Speciﬁcally with regards to the latter, we’ve had to slightly generalize the procedure of [11], in order to derive that β(x) = −V (x)/c−1 + β0.

Thus, in letting y = −x + , → 0, (A.14) becomes

α(x)α0(−x + ) − α(−x + )α0(x) = α()β(x) − β(−x + ), (A.15)

which by a Taylor expansion about = 0 yields

2 2 α(x)α(−x)+α0(−x)+ α00(−x)+O(3)−α0(x)α0(−x)+α00(−x)+ α000(−x)+O(3) 2 2 2 3 = α()β0(x) − β00(x) + β000(x) + O(4), (A.16) 2 6 where we used β(−x) = β(x) (whereby β0(x) is odd and β00(x) even, and so on). On grounds of consistency (see [11]), the asymptotic in zero of α(x) is required to be on the form c α() = −1 + c + c + O(2), (A.17) 0 1

where c−1 6= 0. Then, applying (A.17) to (A.16), implies order for order that d O(1) : c β0(x) = α(x)α0(−x) − α(−x)α0(x) = − α(x)α(−x), (A.18a) −1 dx c O(): c β0(x) − −1 β00(x) = α(x)α00(−x) − α0(−x)α0(x), (A.18b) 0 2 c c O(2): c β0(x) − 0 β00(x) + −1 β000(x) = α(x)α000(−x) − α0(x)α00(−x). (A.18c) 1 2 6 The ﬁrst equation, (A.18a), integrates to

1 1 β(x) = − α(x)α(−x) + β0 = − V (x) + β0 , (A.19) c1 c1

Chapter A 57 Section A.3 A. R. Philip Soliton solutions to Calogero-Moser systems

but if diﬀerentiated, yields

00 0 0 00 0 0 00 c−1β (x) = α (x)α (−x) − α(x)α (−x) + α (−x)α (x) − α(−x)α (x) = 2α0(x)α0(−x) − α(x)α00(−x) + α(x)α00(−x) − α(−x)α00(x), (A.20) which in combination with (A.18b) must imply that 1 c β0(x) = α(x)α00(−x) − α(−x)α00(x). (A.21) 0 2 Consistency with (A.18a) then requires that c −1 α(x)α00(−x) − α(−x)α00(x) = c α(x)α0(−x) − α(−x)α0(x). (A.22) 2 0 Now, like Calogero one might use the Lax function equivalence relations (see §2.4.2) to set c0 = 0. Then (A.22) becomes

α(x)α00(−x) = α(−x)α00(x), (A.23)

which implies directly that α(−x) = −α(x) , i.e. that α(x) is odd.

This is all we require at present. However, in continuing the argument it may be proven that the general form of the Calogero-Moser interparticle potential, consistent with Calogero’s ansatze, is constituted by the elliptic Weierstrass function, ℘(x | ω1, ω2). To this end one should note that (A.18c) simpliﬁes to d c β000(x) + 6c β0(x) = 3α00(x)α0(−x) − α000(x)α(−x) = 3 α00(x)α(−x), (A.24) −1 1 dx which, up to a constant (that may be absorbed into β(x) anyway), integrates to

00 00 c−1β (x) + 6c1β(x) + 3α (x)α(−x) = 0. (A.25)

But from the oddness property of α(x) it follows that

V (x) = −α2(x) ⇒ V 0(x) = −2α0(x)α(x) ⇒ −V 00(x) = 2α00(x)α(x) + 2α0(x)2, whence

2 1 V 0(x) α00(x)α(−x) = 2V 00(x) − . (A.26) 4 V (x)

Hence, using β(x) = −V (x) + const., (A.25) implies

2 3 V 0(x) −c V 00(x) − 6c V (x) + 2V 00(x) − = 0, (A.27) −1 1 4 V (x) or equivalently

00 0 2 2 (6 − 4c−1)V (x)V (x) − 3 V (x) − 24c1 V (x) = 0, (A.28)

which is satisﬁed most generally by the Weierstrass elliptic function, see [11] for details.

Chapter A 58 Section A.3 Appendix B

Appendix on Bäcklund transformations

B.1 Proving the form of the CM Bäcklund Transfor- mations (CMBTs)

We begin by showing a useful lemma, namely

Lemma B.1.1. Let α(x) be the special Lax function to the Calogero-Moser system with V (x) = α(x)α(−x). Let the indices j, k run over the same values. Then X X 0 α (xi − yj) α(xi − yk) − α(yj − yk) = 0 (B.1) j k6=j

Proof. Since j, k run over the same values, it holds for example that

X X 0 X X 0 α (xi − yj)α(xi − yk) = α (xi − yk)α(xi − yj), (B.2) j k6=j j k6=j

whence it is possible to write X X 0 0 α (xi − yj)α(xi − yk) − α(yj − yk)α (xi − yj) j k6=j 1 X X = α0(x − y )α(x − y ) + α0(x − y )α(x − y ) 2 i j i k i k i j j k6=j 0 0 − α(yj − yk)α (xi − yj) − α(yk − yj)α (xi − yk) . (B.3)

However, upon introducing variables x, y such that x = xi − yj, y = yk − xi, and x + y = 0 0 yk −yj, (B.3) must vanish due to the special functional equation, −α (x)α(y)+α (y)α(x)+ α(x + y)α0(x) − α0(y) = 0.

In anticipation of the main proof, it is instructive to consider an additional lemma, essentially due to D.V. Choodnovsky [7]

59 A. R. Philip Soliton solutions to Calogero-Moser systems

Lemma B.1.2. Let α(x) be the special Lax function to the Calogero-Moser system with P V (x) = α(x)α(−x). Then the dynamical equations x˙ i = − j6=i α(xi − xj) imply Hamil- ton’s eqs. of motion

N X 0 X 0 x¨i = − V (xi − xj) = − α (xi − xj) {α(xj − xi) − α(xi − xj)} , (B.4) j6=i j6=i

for the system.

0 0 Proof. Using α(−x) = −α(x) ⇒ α (−x) = α (x), the time-derivative of x˙ i may be written X 0 X 0 X X − α (xi − xj) x˙ i − x˙ j = α (xi − xj) α(xi − xk) − α(xj − xl) , (B.5) j6=i j6=i k6=i l6=j wherein terms with k = j and l = i already contribute X 0 − α (xi − xj) α(xj − xi) − α(xi − xj) =x ¨i, (B.6) j6=i according to (B.4). The remaining terms ought to cancel. These may be gathered into X X 0 α (xi − xj) α(xi − xl) − α(xj − xl) = 0 (B.7) j6=i l6=i l6=j where we used lemma B.1.1 together with the fact that j and l run over the same values.

Lemma B.1.2 simpliﬁes the proof of the following theorem, and components of its proof return here with minor modiﬁcation.

Theorem B.1.1. Let α(x) be a special CM Lax function, and set V (x) = α(x)α(−x). Suppose two systems of complex coordinates {xi ∈ C : i = 1, ..., N} and, respectively, {zi ∈ C : i = 1, ..., N}, are related by the system of coupled equations

 N M  X X  x˙ j = − α(xj − xk) + α(xj − zn) + iωxj,   k6=j n (B.8) M N  X X  z˙ = α(z − z ) − α(z − x ) + iωz ,  n n m n l n  m6=n l where ω = 0 unless V (x) ∝ 1/x. Then the uncoupled dynamical equations

X 0 2 w¨i = − V (wi − wj) − ω wi. (B.9) k6=i are obeyed for wi = xi and wi = zi separately. Therefore (B.8) constitutes a Bäcklund transformation between CM systems of the same potential V (x).

Chapter B 60 Section B.1 A. R. Philip Soliton solutions to Calogero-Moser systems

Proof. We assume ﬁrst that ω = 0. Keep in mind throughout that α(−x) = −α(x) 0 0 implies that α (−x) = α (x). From the time derivative of x˙ i we have that

N M X 0 X 0 x¨i = − α (xi − xj) x˙ i − x˙ j + α (xi − zn) x˙ j − z˙n , (B.10) k6=j n where N N M M X X X X x˙ i − x˙ j = − α(xi − xj) − α(xj − xk) − α(xi − zn) + α(xj − zn) , (B.11) j6=i k6=j n n N M N M X X X X x˙ i − z˙n = − α(xi − xj) + α(xi − zm) − α(zn − zm) + α(zn − xl) .(B.12) j6=i m m6=n l

Lemma B.1.2 shows that x¨i follows from the 1st and 2nd terms of (B.11), whence the remaining terms ought to cancel. In fact, given that N and M are independent, they will cancel separately among two groups of terms. First, we note with respect to (B.12), that for l = i, m = n the 2nd term cancels the 4th. From there we consider the following combination of terms

X X 0 α (xi − zn) α(xi − zm) − α(zn − zm) = 0, (B.13) n m6=n which vanishes virtue of lemma B.1.1, and the fact that indices n, m run over the same values. The remaining terms may be gathered into X X 0 α (xi − xj) − α(xi − zn) + α(xj − zn) n j6=i 0 + α (xi − zn) − α(xi − xj) + α(zn − xj) , (B.14) but in setting x = xi − xj, y = zn − xi, such that x + y = zn − xj, it is found for the terms within the brackets that

α0(x)α(y) − α(x + y) + α0(y) − α(x) + α(x + y) = α0(x)α(y) − α0(y)α(x) − α(x + y)α0(x) − α0(y) = 0. (B.15)

Now, including a ω 6= 0 the derivative of x˙ i acquires the additional terms

N N X 0 X 0 iωx˙ i − iω α (xi − xj) xi − xj + iω α (xi − zn) xi − zn , (B.16) k6=j k6=j which for a rational CM system only, with V (x) ∝ 1/x and α(x) = c−1/x, reduces into

N N X c−1 X c−1 2 iωx˙ i + iω − iω = −ω xi, (B.17) xi − xj xi − zn k6=j k6=j

where the equality follows from the Bäcklund transformation itself. The results for z¨n follows in analog.

Chapter B 61 Section B.2 A. R. Philip Soliton solutions to Calogero-Moser systems

B.2 Derivation of the CMBTs from the Schrödinger equation

The Bäcklund transformations between Calogero Moser systems (CMBTs) may be derived from certain solutions to the Schrödinger equation [22]

iψt = −ψxx + U(z)ψ. (B.18)

If we consider the rational CM Bäcklund transformation, and write φ(x) = 1/x, we should PM 2 set U(z) = 2 n φ (z − zn) and seek rational solutions on the form

N iε(λz−λ2t) Πj (z − xj) κ X ψ(z, t) = e M := e . (B.19) Πn (z − zn) Y If follows that X X X ψ = eκ − iλ2 − x˙ φ(z − x ) + z˙ φ(z − z ) , (B.20) t Y j j n n j n

and X X U(z)ψ(z) = 2 φ2(z − z )eκ . (B.21) n Y n Furthermore X X X X ψ = eκ iλ + φ(z − x ) − φ(z − y ) := eκ Z , (B.22) z Y j j Y j n whereby

X X X ψ = eκ Z2 − φ2(z − x ) + φ2(z − z ) . (B.23) zz Y j n j n

Combining the terms we ﬁnd

X X X iψ + ψ − U(z)ψ = eκ λ2 − i x˙ φ(z − x ) + i z˙ φ(z − z ) (B.24) t xx Y j j n n j n 2 X 2 X 2 X 2 + Z − φ (z − xj) + φ (z − zn) − 2 φ (z − zn) = 0 (B.25) j n n

where 2 2 2 X X Z = − λ + 2iλ φ(z − xk) − φ(z − zm) (B.26) k m 2 2 X X X X + φ(z − xk) − 2 φ(z − xk)φ(x − zm) + φ(z − zm) k k m z (B.27)

Chapter B 62 Section B.2 A. R. Philip Soliton solutions to Calogero-Moser systems

Note that for ± 1 we have that λ2 − 2λ2 = 0. Removing these and other terms that cancel we ﬁnd

X X X X − i x˙ jφ(z − xj) + i z˙nφ(z − zn) + 2iλ φ(z − xk) − φ(z − zm) j n k m X X X X + φ(z − xj)φ(z − xk) − 2 φ(z − xj)φ(z − zn) j k6=j j n X X + φ(z − zn)φ(z − zm) = 0 (B.28) n m6=n which upon taking residues at z = xj and z = zn respectively, implies that X X −ix˙ j = φ(xj − xk) − φ(xj − zn) − 2iλ, (B.29) k6=j n X X iz˙n = φ(zn − xk) − φ(zn − zm) + 2iλ. (B.30) k m6=n

Chapter B 63 Section B.2 Appendix C

Appendix on hydrodynamics and solitons

C.1 Some facts about the Hilbert transform

The Hilbert transform over the real line of a function f(z), z ∈ C is given by

p.v. Z ∞ f(z) H[ f ](x) = dz, (C.1) π −∞ x − z if the Cauchy principal value (p.v.) of the integral exists. For periodic functions there is also the Hilbert transform over the circle with period L

p.v. Z L π H[ f ](x) = f(z) cot (x − z)dz. (C.2) L 0 L If the function is analytic in only the upper half plane C+ (or outside the complex circle with period L) or the lower half plane C− (or inside the circle) it holds that [2]

H[ f ](x) = ∓if ±(x). (C.3)

Note also that the transform commutes with derivatives

0 ∂xH[ f ](x) = H[ f ](x). (C.4)

C.2 Proof of theorem 2.7.1

We begin with a useful lemma

Lemma C.2.1. Let α(x) be the special Lax function to the Calogero-Moser system with V (x) = α(x)α(−x). Then it holds that

0 0 0 0 α(z − xi)α (z − yn) + α(z − yn)α (z − xi) = α(yn − xi) α (z − yn) − α (z − xi) (C.5)

Proof. Introducing new variables x, y such that x = z − xi, y = yn − z, and using α0(x) = α0(−x) while α(x) = −α(−x)), we may write

0 0 0 0 α(z − xi)α (z − yn) + α(z − yn)α (z − xi) = α(x)α (y) − α(y)α (x) (C.6)

64 A. R. Philip Soliton solutions to Calogero-Moser systems

However, by the special functional equation

α(x)α0(y) − α(y)α0(x) = α(x + y)α0(y) − α0(x) (C.7) 0 0 = α(yn − xi) α (z − yn) − α (z − xi) . (C.8)

We can now prove the following

Theorem C.2.1. To the BH equation, ut = 2cuux + c−1cuxx, a class of solutions is constituted by the meromorphic functions

N X u(x, t) = α(x − xj) − iωx , (C.9) j for arbitrary N, if and only if their poles, xj : j = 1, ..., N , satisfy

N X x˙ j = −2c α(xj − xk) + 2ciωxj. (C.10) k

PN 0 Proof. We consider ﬁrst the case ω = 0. (C.9) implies that ut = − i α (x − xi)x ˙ i and

X 00 X 2 c−1cuxx = c−1c α (x − xi) = −c ∂xα (x − xi), (C.11) i i

0 2 where we used α (x) = −α (x)/c−1 + const.. The remaining term

N X 0 2cuux = 2c α(x − xi)α (x − xj), (C.12) ij for i = j contributes

X 0 X 2 2c α(x − xi)α (x − xi) = c ∂xα (x − xi) = −c−1cuxx, (C.13) i i whence it cancels the RHS of (C.11). The remainder may be written as

X X 0 2c α(x − xi) α (x − xj) i j6=i X X 0 0 = c α(x − xi)α (x − xj) + α(x − xj)α (x − xi) i j6=i X 0 X = 2c α (x − xi) α(xi − xj), (C.14) i j6=i where the last step follows from lemma C.2.1. We conclude that the non-vanishing components of the BH equation combine to make

N X 0 X 0 X − α (x − xi)x ˙ i = 2c α (x − xi) α(xi − xj), (C.15) i i j6=i

Chapter C 65 Section C.2 A. R. Philip Soliton solutions to Calogero-Moser systems

P with the residue x˙ i = −2c j6=i α(xi − xj) as x → xi. Now if ω 6= 0, the 2cuux acquires additional terms X X 0 −2iωc α(x − xj) + x α (x − xk) − iωx . (C.16) j k The ﬁrst term diverges too slowly to contribute to the residue, and the third term does not diverge at all. However, the second term contributes a term −2ciωxi to the residue calculation (C.15). Hence we have that X x˙ i = −2c α(xi − xj) + 2ciωxi (C.17) j6=i

C.3 Proof of theorem 2.7.2

Theorem C.3.1. The meromorphic functions

N M X X u(x, t) = α(x − xi) − α(x − zn) − iωx, (C.18) i n

in special Lax functions α(x) with null asymptotic c−1 = i, for arbitrary N and M, form a class of solutions to the BO equation, ut + 2cuux + cH [uxx] = 0, if and only if their poles, xi : i = 1, ..., N and zn : n = 1, ..., M , satisfy the system of equations

 N M  X X  − x˙ j = 2c α(xj − xk) − 2c α(xj − zn) − 2ciωxj,   k6=j n (C.19) N M  X X  z˙ = 2c α(z − z ) − 2c α(z − x ) + 2ciωz .  n n m n l n  m6=n l

PN 0 Proof. First we prove the case with ω = 0. We have that ut = − i α (x − xi)x ˙ i + PM 0 n α (x − zn)z ˙n. Assuming that the two sets of poles inhabit separate halves of the imaginary plane (one set above the real line and one set below, see [8] and appendix C.1) the Hilbert transform yields

N M X 00 X 00 cH[uxx] = c−1c α (x − xi) + α (x − zn) , (C.20) i n since c−1 = i. Consider the remaining term

N M N M X X X 0 X 0 2cuux = 2c α(x − xi) − α(x − zn) α (x − xj) − α (x − zm) i n j m X 0 X 0 = 2c α(x − xi)α (x − xj) − α(x − xi)α (x − zm) ij im X 0 X 0 − α(x − zn)α (x − xj) + α(x − zn)α (x − zm) . (C.21) jn nm

Chapter C 66 Section C.3 A. R. Philip Soliton solutions to Calogero-Moser systems

From the proof of the theorem in appendix C.2, concerning the Burger’s Hopf equation, we know that the terms pure in xj or zn should yield X 0 X X 00 2c α (x − wi) α(wi − wj) − c−1c α (x − wi), (C.22) i j6=i i for wi = xi and wi = zi respectively. The mixed terms can be written X 0 0 − 2c α(x − xi)α (x − zn) + α(x − zn)α (x − xi) in X 0 0 = −2c α(xi − zn)α (x − xi) + α(zn − xi)α (x − zn) , (C.23) in where the RHS follows from lemma C.2.1 of appendix C.2. Putting everything together the BO equation becomes

N M X 0 X 0 − α (x − xi)x ˙ i + α (x − zn)z ˙n = i n X 0 X X 0 X 2c α (x − xi) α(xi − xj) + 2c α (x − zn) α(zn − zm) i j6=i n m6=n X 0 X X 0 X − 2c α (x − xi) α(xi − zn) − 2c α (x − zn) α(zn − xi), (C.24) i n n i and a residue calculation for x → xi and x → zn respectively, then immediately yields the system of equations  N M  X X  − x˙ j = 2c α(xj − xk) − 2c α(xj − zn),   k6=j n=1 (C.25) M N  X X  z˙ = 2c α(z − z ) − 2c α(z − x ).  n n m n k  m6=n k Now, to include a harmonic external potential into the residue equations, we should consider solutions on the form N M X X u(x, t) = α(x − xi) − α(x − zn) − iωx. (C.26) i n

Then the BO equation, from 2cuux, acquires the additional terms X 0 X 0 X X − 2ciω x α (x − xi) − α (x − zn) + α(x − xi) − α(x − zn) − iωx . i n i n (C.27) The three ﬁnal terms diverge too slowly to contribute to the residues (the last one not at all), while the ﬁrst two terms contribute to make  N M  X X  − x˙ j = 2c α(xj − xk) − 2c α(xj − zn) − 2icωxj,   k6=j n (C.28) N M  X X  z˙ = 2c α(z − z ) − 2c α(z − x ) + 2icωz ,  n n m n l n  m6=n l as x → xi and x → zn respectively.

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Chapter C 70 Section C.3