Integrable Systems

4Spectral Methods for Solving Integrable Systems

The aim of this chapter is to present an overview of the active area via the spectral linearization methods for solving integrable systems. New examples of completely integrable Hamiltonian systems, which have been discovered, are based on the so-called Lax representation (Lax pairs) of the equations of motion. We will explain how these systems can be realized as straight line motions on a Jacobi variety of a spectral curve. These methods are exemplified by several problems of integrable systems of relevance in mathematics and mathematical physics: we study the geodesic flow on the group SO(n) and more particularly, on SO(4). We are also interested in the study of Euler, Lagrange, Kowalewski and Goryachev–Chaplygin spinning tops, Jacobi geodesic flow on an ellipsoid and the Neumann problem. Other important examples include a family of integrable systems, the coupled nonlinear Schrödinger equations, the Yang–Mills equations and the periodic infinite band matrix.

4.1. Lax equations and spectral curves

A Lax equation is given by a differential equation of the form

[4.1]

where are functions depending on a parameter h (spectral parameter), whose coefficients A_k and B_k are matrices in Lie algebras. The pair (A, B) is called a Lax pair. ...