Selected Topics in Classical Integrability

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. ∗This article is based on a series of lectures presented at the University of Oldenburg in July 2011.