Selected Topics in Classical Integrability

TL;DR

This paper reviews fundamental concepts of classical integrable systems, including algebraic descriptions, zero curvature conditions, and the classical r-matrix approach, with examples like Toda chain and sine-Gordon models.

Contribution

It provides a comprehensive overview of algebraic methods and key models in classical integrability, emphasizing the r-matrix framework and Lax pairs.

Findings

Algebraic description of integrable models

Derivation of integrals of motion and Lax pairs

Discussion of discrete and continuum models

Abstract

Basic notions regarding classical integrable systems are reviewed. An algebraic description of the classical integrable models together with the zero curvature condition description is presented. The classical r-matrix approach for discrete and continuum classical integrable models is introduced. Using this framework the associated classical integrals of motion and the corresponding Lax pair are extracted based on algebraic considerations. Our attention is restricted to classical discrete and continuum integrable systems with periodic boundary conditions. Typical examples of discrete (Toda chain, discrete NLS model) and continuum integrable models (NLS, sine-Gordon models and affine Toda field theories) are also discussed.