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Integrable Systems and Their Finite-Dimensional Reductions

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Integrable Systems and Their Finite-Dimensional Reductions Integrable Systems and their Finite-Dimensional Reductions Andrew N. W. Hone Doctor of Philosophy University of Edinburgh 1996 To my mother, Louise Hone. Abstract This thesis concerns some of the different ways in which integrable systems ad- mit finite-dimensional reductions. The point is that partial diferential equations which are integrable (usually in the sense of being solvable by inverse scatter- ing) admit certain special classes of solutions, and these special solutions may be viewed as finite-dimensional mechanical systems in their own right. structures introduces some of the important concepts and The first chapter grability, and includes a brief overview of some of the appli- associated with inte cations of integrable systems and their reductions in field theory. Chapter 2 describes the scaling similarity reductions of the SawadaKotera, Kuper5hmdt equations. similarity solutions of these fifth-order KdV, and Kaup s which are naturally viewed as fourth- evolution equations satisfy certain ODE ranscendents; they may also be written as non- order analogues of the Painlevé t generalizations of autonomous Hamiltonian systems, which are time-dependent the integrable Henon-Heiles systems. The solutions to these systems are encoded into a taufuncti0n, and Bäcklund transformations are presented which allow the solutions and some other special solutions. construction of rational solu- concerned with the motion of the poles of singular The third chapter is emonstrated that ional solutions)on of the NLS equation. It is d tions (especially rat well-known Crum trans- the linear problem for NLS admits an analogue of the construction of a sequence of hrodinger operators leading to the formation for S c rational solutions. The poles and zeros of these rational solutions are found to satisfy constrained Calogero.M05er equations, and some other singular solutions are also considered. Much use is made of ilirota's bilinear formalism, as well as a trilinear form for NLS related to its reduction from the KP hierarchy. AV affine Toda field The final chapter deals with soliton solutions of the theories. By writing the soliton tau-functions as determinants of a particular form, these solutions are related to the hyperbolic spin RuijsenaarsSchneider system. These results generalize the connection between the ordinary (non-spin) Ruijsenaarseer model and the soliton solutions of the sine-Gordon equa- tion. Acknowledgements This thesis is based on research carried out in the Department of Mathematics and Statistics of the University of Edinburgh in the years 1993-1996, funded by the EPSRC. I would like to take the opportunity to thank some of the people who have made it possible. First of all, I am very grateful to my supervisor Harry Braden for introduc- ing me to the subject and steering me in the right direction, as well as for his constant encouragement. He has been an indispensable source of knowledge and experience, and has shown great patience in lending a critical ear to my ideas and suggestions, however unfeasible. Also he has been extremely tolerant of my frequent failure to talk maths before coffee time. Many thanks are due to the staff and research students in the Edinburgh Mathematics and Physics Departments, for all the stimulating lectures, seminars and discussions. I have also benefited a great deal from talking to the people I have met at various conferences over the past three years. In particular, I would like to thank Allan Fordy for encouraging me to develop the material in Chapter 2, and Orlando Ragnisco for showing interest in some of my other work. I must also give a special mention to the staff and students at the Institute of Theoretical Physics of Warsaw University, who were very hospitable during the School of Nonlinearity and Geometry. The greatest thanks go to my parents for all their love and support. They have always given me the freedom to think for myself, and to them I owe my love of learning. It is a pleasure to thank all my family and friends, who have helped to make life so fun and exciting. Extra special thanks to Ali and Jasper for all the times they've been there for me, and to Dave for putting up with my frequent descents into incomprehensibility. Above all, lam grateful to Jemma for her constant love and artistic inspiration. Declaration I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text. (Andrew N. W. Hone) Table of Contents Chapter 1 Introduction 1 1.1 General Introduction ......................... 1 1.2 Integrable Hamiltonian Systems in Finite Dimensions ....... 3 1.2.1 Finite-Dimensional Hamiltonian Mechanics ......... 4 1.2.2 Lax Pairs ........................... 6 1.2.3 The Calogero-Moser System ................. 6 1.3 The KdV Hierarchy ........................... 11 1.3.1 Hamiltonian Formulation of KdV .............. 12 1.3.2 Lax Formalism and Inverse Scattering ........... 15 1.3.3 Miura Map and Bãcklund Transformations ......... 19 1.3.4 Bilinear Form and Hirota's Method ............. 21 1.4 The KP Hierarchy .......................... 23 1.4.1 The Lax Operator for KP .................. 24 1.4.2 Sato Theory and the Tau-Function of KP .......... 26 1.5 Reductions of Integrable Systems in Field Theory ......... 29 1.5.1 Solitons and Field Theory .................. 29 1.5.2 Random Matrices, Correlation Functions and the Painlevé Transcendents ......................... 32 1.5.3 Matrix Models of 2-D Quantum Gravity .......... 36 Chapter 2 Non-autonomous Hénon-Heiles Systems 41 2.1 Introduction .............................. 41 2.2 Scaling Similarity Solutions in the Sawada-Kotera, KdV and Kaup- Kupershmidt Hierarchies ....................... 47 2.2.1 General Description of Scaling Similarity Solutions . 48 2.2.2 Similarity Solutions of KdV and PIT ............ 52 2.2.3 Similarity Solutions of Fifth-Order Equations ....... 52 2.3 Zero Curvature Equations ...................... 54 2.4 Hénon-Heiles Systems ......................... 58 2.4.1 Stationary Flows and Integrable Hénon-Heiles ....... 58 2.4.2 Non-autonomous Hamiltonians for Scaling Similarity Solu- 58 tions ............................ 62 2.4.3 The Hamiltonian System fl(ii) ................ 67 2.4.4 The Hamiltonian Systems flj and 'H(iii) .......... 70 2.4.5 Analogues of the Toda Lattice for Sequences of Tau-Functions 72 2.5 Painlevé Analysis and Special Solutions ............... 72 2.5.1 The Painlevé Test ....................... 73 2.5.2 Rational Solutions ...................... 77 2.5.3 Solutions Related to P1 .................... 79 2.6 Generalizations ........................ 79 2.6.1 A Conjecture ......................... ................. 79 2.6.2 More ilenon-Heiles Systems 80 2.6.3 Scaling Similarity Solutions to the Hirota-Satsuma System 81 2.7 Conclusion ......................... Chapter 3 Singular Solutions of the Nonlinear Schrödinger Equa- 83 tion and their Pole Dynamics 83 3.1 Introduction ....................... ................... 83 3.1.1 KP and Calogero-Moser 85 3.1.2 Pole Motion for NLS ..................... 86 3.2 NLS and the Singular Manifold Method ............... 87 3.2.1 NLS and AKNS ........................ 88 3.2.2 Inverse Scattering and the ABT ............... ....... 89 3.2.3 Application of the Singular Manifold Method 92 3.3 NLS and its Auto-Bäcklund Transformation in Bilinear Form 92 3.3.1 The Bilinear Form ...................... ........ 93 3.3.2 Bãcklund Transformations in Bilinear Form ................ 94 3.3.3 Classical Boussinesq and NLS 96 3.3.4 Bilinear ABT ....................... 98 3.4 The NLS Crum Transformation and Rational Solutions ...... 98 3.4.1 The Crum Transformation for KdV ............. 99 3.4.2 NLS Crum Transformation .................. 100 3.4.3 A Sequence of Rational Functions .............. ..................... 105 3.4.4 Wronskian Formulae 110 3.5 The Trilinear Form of NLS ...................... ............. 1 10 3.5.1 Direct Derivation of Trilinear Form 112 3.5.2 AKNS as a Reduction of KP ................. ...... 115 3.6 Dynamics of the Poles and Zeros of Rational Solutions 11 3.6.1 Calogero-Moser via the Trilinear Form 116 3.6.2 Coupled Equations for Poles and Zeros ........... 117 3.7 Similarity Solutions and other Singular Solutions ......... 119 3.7.1 Scaling Similarity Solutions ................. 120 3.7.2 Two-Lump Solution ...................... 124 3.8 Conclusions .............................. 126 3.9 Appendix A .............................. 128 3.10 Appendix B .............................. 129 Chapter 4 Affine Toda Solitons and Ruijsenaars-Schneider Systems with Spin 134 4.1 Introduction .............................. 134 4.2 Affine Toda Field Theories ...................... 136 41) 4.3 The Affine Toda Solitons .................... 138 4.3.1 Soliton Tau-Functions .................... 138 4.3.2 ABT and Soliton Determinants ............... 139 4.4 The Reduced Symplectic Form .................... 142 4.5 Ruijsenaars-Schneider Systems .................... 143 4.5.1 Spin RS Model ........................ 143 4.5.2 Spin RS Equations from Toda Solitons ........... 144 4.5.3 Sine-Gordon and Spinless RS ................ 146 4.6 Conclusion ............................... 147 Bibliography 149 111 Chapter 1 Introduction 1.1 General Introduction Whenever one considers the various models available to describe the physical nreasonable effectiveness of mathematics". world, one is often struck by "the u General relativity and quantum mechanics, the two main pillars of twentieth cen-
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