Finite dimensional reductions of integrable differential-difference equations

Abstract: Integrable partial differential equations, such as the Korteweg-de Vries (KdV) equation, admit infinite hierarchies of commuting higher symmetries. Their symmetry reductions give rise to integrable finite-dimensional dynamical systems that are solvable in terms of Abelian functions. This observation underlies the finite-gap integration method for the KdV equation, introduced by S.P. Novikov and subsequently extended to a wide class of integrable systems. In this paper, we introduce a new and more general class of reductions for integrable differential-difference equations, leading to integrable finite-dimensional systems in both commutative and noncommutative settings. The reduction constraints define integrable maps that enable solutions of the reduced systems to be extended to solutions of the corresponding differential-difference equations. The construction is illustrated using the Volterra and Toda hierarchies.

mathematical physicsanalysis of PDEsdifferential geometry

Audience: researchers in the topic

Geometry of differential equations seminar