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On the symmetry classification of integrable chains in 3D. Darboux-integrable reductions and their higher symmetries

R. N. Garifullin, I. T. Habibullin

TL;DR

This work addresses the classification of integrable nonlinear equations with three independent variables by leveraging Darboux-integrable open-chain reductions as a symmetry-based filter. It shows that all known integrable Toda-type lattices admit length-three open-chain reductions that possess second-order evolutionary symmetries, providing a concrete, practically applicable criterion for identifying integrable $3$D differential-difference equations. The approach offers a simpler alternative to the traditional, algebraically heavy Darboux integrability tests and yields a constructive pathway to obtain symmetry hierarchies for the reductions. The results have potential to streamline the cataloging of integrable 3D chains and to facilitate the discovery of new integrable models by focusing on symmetry-rich open-chain reductions.

Abstract

This paper proposes a method for identifying and classifying integrable nonlinear equations with three independent variables, one of which is discrete and the other two are continuous. A characteristic property of this class of equations, called Toda-type chains, is that they admit finite-field reductions in the form of open chains with enhanced integrability. The paper results in a theorem stating that all known integrable Toda-type chains admit reductions in the form of an open chain of length three with a family of second-order evolutionary type symmetries. Apparently, this property of Toda-type chains can be used as an effective classification criterion when compiling lists of integrable differential-difference equations in 3D.

On the symmetry classification of integrable chains in 3D. Darboux-integrable reductions and their higher symmetries

TL;DR

This work addresses the classification of integrable nonlinear equations with three independent variables by leveraging Darboux-integrable open-chain reductions as a symmetry-based filter. It shows that all known integrable Toda-type lattices admit length-three open-chain reductions that possess second-order evolutionary symmetries, providing a concrete, practically applicable criterion for identifying integrable D differential-difference equations. The approach offers a simpler alternative to the traditional, algebraically heavy Darboux integrability tests and yields a constructive pathway to obtain symmetry hierarchies for the reductions. The results have potential to streamline the cataloging of integrable 3D chains and to facilitate the discovery of new integrable models by focusing on symmetry-rich open-chain reductions.

Abstract

This paper proposes a method for identifying and classifying integrable nonlinear equations with three independent variables, one of which is discrete and the other two are continuous. A characteristic property of this class of equations, called Toda-type chains, is that they admit finite-field reductions in the form of open chains with enhanced integrability. The paper results in a theorem stating that all known integrable Toda-type chains admit reductions in the form of an open chain of length three with a family of second-order evolutionary type symmetries. Apparently, this property of Toda-type chains can be used as an effective classification criterion when compiling lists of integrable differential-difference equations in 3D.
Paper Structure (4 sections, 55 equations)

This paper contains 4 sections, 55 equations.

Table of Contents

  1. Introduction
  2. Special (locking) boundary conditions for integrable 3D models and open chain type reductions.
  3. Symmetries of open chains.
  4. Conclusions.

Theorems & Definitions (1)

  • proof