On Separation of Variables for Integrable Equations of Soliton Type

TL;DR

This paper introduces a general scheme for separation of variables in integrable Hamiltonian systems, demonstrated on several well-known soliton equations, enhancing understanding and solution methods for these complex systems.

Contribution

It presents a novel, general scheme for variable separation in integrable systems on loop algebra orbits, applied to key soliton equations.

Findings

Scheme successfully applied to MKdV, sine-Gordon, NLS, and Heisenberg magnetic equations.

Provides a unified approach to separation of variables in integrable systems.

Enhances analytical tools for solving soliton equations.

Abstract

We propose a general scheme for separation of variables in the integrable Hamiltonian systems on orbits of the loop algebra $\mathfrak{sl}(2,\Complex)\times \mathcal{P}(\lambda,\lambda^{-1})$. In particular, we illustrate the scheme by application to modified Korteweg--de Vries (MKdV), sin(sinh)-Gordon, nonlinear Schr\"odinger, and Heisenberg magnetic equations.