This book illustrates the powerful interplay between topological, algebraic and complex analytical methods, within the field of integrable systems, by addressing several theoretical and practical aspects. Contemporary integrability results, discovered in the last few decades, are used within different areas of mathematics and physics. Integrable Systems incorporates numerous concrete examples and exercises, and covers a wealth of essential material, using a concise yet instructive approach. This book is intended for a broad audience, ranging from mathematicians and physicists to students pursuing graduate, Masters or further degrees in mathematics and mathematical physics. It also serves as an excellent guide to more advanced and detailed reading in this fundamental area of both classical and contemporary mathematics.

Table of Contents:
Preface ix Chapter 1 Symplectic Manifolds 1 1.1 Introduction 1 1.2 Symplectic vector spaces 2 1.3 Symplectic manifolds 3 1.4 Vectors fields and flows 7 1.5 The Darboux theorem 22 1.6 Poisson brackets and Hamiltonian systems 25 1.7 Examples 31 1.8 Coadjoint orbits and their symplectic structures 35 1.9 Application to the group SO(n) 37 1.9.1 Application to the group SO(3) 39 1.9.2 Application to the group SO(4) 41 1.10 Exercises 47 Chapter 2 Hamilton–Jacobi Theory 49 2.1 Euler–Lagrange equation 49 2.2 Legendre transformation 52 2.3 Hamilton’s canonical equations 53 2.4 Canonical transformations 55 2.5 Hamilton–Jacobi equation 57 2.6 Applications 60 2.6.1 Harmonic oscillator 60 2.6.2 The Kepler problem 61 2.6.3 Simple pendulum 62 2.7 Exercises 64 Chapter 3 Integrable Systems 67 3.1 Hamiltonian systems and Arnold–Liouville theorem 67 3.2 Rotation of a rigid body about a fixed point 75 3.2.1 The Euler problem of a rigid body 78 3.2.2 The Lagrange top 82 3.2.3 The Kowalewski spinning top 83 3.2.4 Special cases 90 3.3 Motion of a solid through ideal fluid 91 3.3.1 Clebsch’s case 91 3.3.2 Lyapunov–Steklov’s case 93 3.4 Yang–Mills field with gauge group SU(2) 93 3.5 Appendix (geodesic flow and Euler–Arnold equations) 95 3.6 Exercises 100 Chapter 4 Spectral Methods for Solving Integrable Systems 103 4.1 Lax equations and spectral curves 103 4.2 Integrable systems and Kac–Moody Lie algebras 104 4.3 Geodesic flow on SO(n) 107 4.4 The Euler problem of a rigid body 108 4.5 The Manakov geodesic flow on the group SO(4) 109 4.6 Jacobi geodesic flow on an ellipsoid and Neumann problem 114 4.7 The Lagrange top 115 4.8 Quartic potential, Garnier system 115 4.9 The coupled nonlinear Schrödinger equations 118 4.10 The Yang–Mills equations 119 4.11 The Kowalewski top 119 4.12 The Goryachev–Chaplygin top 121 4.13 Periodic infinite band matrix 122 4.14 Exercises 122 Chapter 5 The Spectrum of Jacobi Matrices and Algebraic Curves 129 5.1 Jacobi matrices and algebraic curves 129 5.2 Difference operators 135 5.3 Continued fraction, orthogonal polynomials and Abelian integrals 137 5.4 Exercises 140 Chapter 6 Griffiths Linearization Flows on Jacobians 143 6.1 Spectral curves 143 6.2 Cohomological deformation theory 144 6.3 Mittag–Leffler problem 148 6.4 Linearizing flows 149 6.5 The Toda lattice 150 6.6 The Lagrange top 153 6.7 Nahm’s equations 154 6.8 The n-dimensional rigid body 155 6.9 Exercises 156 Chapter 7 Algebraically Integrable Systems 159 7.1 Meromorphic solutions 159 7.2 Algebraic complete integrability 164 7.3 The Liouville–Arnold–Adler–van Moerbeke theorem 171 7.4 The Euler problem of a rigid body 173 7.5 The Kowalewski top 175 7.6 The Hénon–Heiles system 191 7.7 The Manakov geodesic flow on the group SO(4) 200 7.8 Geodesic flow on SO(4) with a quartic invariant 206 7.9 The geodesic flow on SO(n) for a left invariant metric 210 7.10 The periodic five-particle Kac–van Moerbeke lattice 212 7.11 Generalized periodic Toda systems 213 7.12 The Gross–Neveu system 214 7.13 The Kolossof potential 215 7.14 Exercises 215 Chapter 8 Generalized Algebraic Completely Integrable Systems 221 8.1 Generalities 221 8.2 The RDG potential and a five-dimensional system 225 8.3 The Hénon–Heiles problem and a five-dimensional system 229 8.4 The Goryachev–Chaplygin top and a seven-dimensional system 231 8.5 The Lagrange top 236 8.6 Exercises 237 Chapter 9 The Korteweg–de Vries Equation 241 9.1 Historical aspects and introduction 241 9.2 Stationary Schrödinger and integral Gelfand–Levitan equations 243 9.3 The inverse scattering method 255 9.4 Exercises 269 Chapter 10 KP–KdV Hierarchy and Pseudo-differential Operators 275 10.1 Pseudo-differential operators and symplectic structures 275 10.2 KdV equation, Heisenberg and Virasoro algebras 279 10.3 KP hierarchy and vertex operators 281 10.4 Exercises 290 References 293 Index 305