Symmetry algebra of discrete KdV equations and corresponding differential-difference equations of Volterra type

TL;DR

This paper explores the symmetry algebra of discrete KdV equations, deriving conservation laws and hierarchies of symmetries, revealing Virasoro algebra structures and linking them to differential-difference equations.

Contribution

It introduces a comprehensive symmetry algebra framework for discrete KdV equations and related equations, including explicit structures and hierarchies of conservation laws.

Findings

Symmetry algebras form Virasoro type structures.

Explicit symmetry algebra for discrete potential KdV.

Hierarchies of zero curvature representations derived.

Abstract

A sequence of canonical conservation laws for all the Adler-Bobenko-Suris equations is derived and is employed in the construction of a hierarchy of master symmetries for equations H1-H3, Q1-Q3. For the discrete potential and Schwarzian KdV equations it is shown that their local generalized symmetries and non-local master symmetries in each lattice direction form centerless Virasoro type algebras. In particular, for the discrete potential KdV, the structure of its symmetry algebra is explicitly given. Interpreting the hierarchies of symmetries of equations H1-H3, Q1-Q3 as differential-difference equations of Yamilov's discretization of Krichever-Novikov equation, corresponding hierarchies of isospectral and non-isospectral zero curvature representations are derived for all of them.