Formal diagonalization of the discrete Lax operators and construction of conserved densities and symmetries for dynamical systems

TL;DR

This paper introduces a new method for formal diagonalization of discrete Lax operators, enabling the calculation of conservation laws and symmetries for various discrete dynamical systems, with applications to several integrable models.

Contribution

It proposes an alternative approach to formal diagonalization of discrete Lax operators, facilitating the derivation of conservation laws and symmetries in discrete integrable systems.

Findings

Derived infinite conservation laws for Toda lattice on a quad graph.

Constructed conservation laws for discrete potential KdV and lattice derivative NLS.

Represented systems including lattice versions of matrix and vector NLS equations.

Abstract

An alternative method of constructing the formal diagonalization for the discrete Lax operators is proposed which can be used to calculate conservation laws and in some cases generalized symmetries for discrete dynamical systems. Discrete potential KdV equation, lattice derivative nonlinear Schr\"odinger equation, dressing chain, Toda lattice are considered as illustrative examples. For the Toda lattice on a quad graph corresponding to the Lie algebra $A_1^{(1)}$ infinite series of conservation laws are described. Systems of quad graph equations are represented including lattice versions of the "matrix" NLS and "vector" derivative NLS equations.