Asymptotic diagonalization of the Discrete Lax pair around singularities and conservation laws for dynamical systems

TL;DR

This paper develops a formal diagonalization method for discrete Lax operators to derive conservation laws and symmetries in nonlinear quad systems, including new models and classical equations.

Contribution

It introduces a novel asymptotic diagonalization technique for discrete Lax operators, enabling effective analysis of conservation laws and symmetries in integrable systems.

Findings

Constructed asymptotic eigenfunction representations.

Derived infinite series of conservation laws for specific systems.

Identified conservation laws and symmetries for a new multiquadratic model.

Abstract

A method of the formal diagonalization of the discrete linear operator with a parameter is studied. In the case when the operator provides a Lax operator for a nonlinear quad system the formal diagonalization method allows one to describe effectively conservation laws and generalized symmetries for this system. Asymptotic representation of the Lax operators eigenfunctions are constructed and infinite series of conservation laws are described for the quad system connected with $A^{(1)}_3$ affine Lie algebra, for the modified discrete Boussinesq system and for the discrete Tzitzeica equation. For a newly found multiquadratic discrete model conservation laws and several generalized symmetries are presented.