Characteristic Lie Algebras of Integrable Differential-Difference Equations in 3D

Abstract

The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward the following hypothesis. Any integrable equation of the type under consideration admits an infinite sequence of finite-field Darboux-integrable reductions. The property of Darboux integrability of a finite-field system is formalized as finite-dimensionality condition of its characteristic Lie-Rinehart algebras. That allows one to derive effective integrability conditions in the form of differential equations on the right hand side of the equation under study. To test the hypothesis, we use known integrable equations from the class under consideration. In this article, we show that all known examples do have this property.

arXiv comment24 pages

References in corpus (17)

• Einstein-Weyl geometry, the dKP equation and twistor theory
• Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability
• Integrable Background Geometries
• On Darboux integrability of discrete 2D Toda lattices
• On a classification algorithm of the integrable two-dimensional lattices via Lie-Rinehart algebras
• Characteristic Lie rings, finitely-generated modules and integrability conditions for 2+1 dimensional lattices
• On dispersionless BKP hierarchy and its reductions
• Dispersive deformations of hydrodynamic reductions of 2D dispersionless integrable systems
• On the Classification of Darboux Integrable Chains
• Complete list of Darboux Integrable Chains of the form $t_{1x}=t_x+d(t,t_1)$
• Classification of a Subclass of Two-Dimensional Lattices via Characteristic Lie Rings
• Integrable equations in 2+1-dimensions: deformations of dispersionless limits
• On Darboux Integrable Semi-Discrete Chains
• On a class of 2D integrable lattice equations
• Cartan matrices and integrable lattice Toda field equations
• Dunajski-Tod equation and reductions of the generalized dispersionless 2DTL hierarchy
• Dispersionless Hirota equations and the genus 3 hyperelliptic divisor

Cited by in corpus (1)

• An algebraic criterion of the Darboux integrability of differential-difference equations and systems