Algebraic constructions of integrable dynamical systems-extensions of the Volterra system

Abstract

CONTENTS Introduction Chapter I. Integrable discretizations of the Korteweg-de Vries equation § 1. Integrable dynamical systems with quadratic non-linearity § 2. Integrable reductions of dynamical systems (1.3) § 3. Inregrable dynamical systems with an arbitrary degree of non-linearity Chapter II. The integrable integro-differential equation § 1. The integro-differential equation as a continuous limit of the family of dynamical systems § 2. The basic properties of the integro-differential equation (1.3) Chapter III. Integrable differential equations in algebras of smooth functions and in continuous associative algebras § 1. First integrals of differential equations connected with automorphisms of associative algebras § 2. Algebraic constructions of certain integrable equations § 3. Differential and integro-differential equations in algebras of functions § 4. A theorem on two commuting automorphisms and its applications § 5. Applications to the Euler equations on the direct sums of Lie algebras and Chapter IV. Integrable dynamical systems connected with simple Lie algebras § 1. Algebraic analogues of the Volterra system § 2. Integrable Hamiltonian perturbations of the generalized Toda lattices § 3. Differential equations admitting Lax representations with several spectral parameters § 4. Dynamical systems admitting a Lax representation and generalizing the Toda lattice References