Symmetries of the Continuous and Discrete Krichever-Novikov Equation

TL;DR

This paper classifies symmetries of a class of differential-difference equations, identifying integrable cases with higher symmetry dimensions and providing insights into their algebraic structure.

Contribution

It performs a symmetry classification for a broad class of differential-difference equations, highlighting the integrable discretizations of the Krichever-Novikov equation.

Findings

Maximum symmetry dimension is 5, occurring only in integrable cases.

A 6-parameter subclass corresponds to an integrable discretization.

Symmetry algebra dimension ranges from 1 to 5 across the class.

Abstract

A symmetry classification is performed for a class of differential-difference equations depending on 9 parameters. A 6-parameter subclass of these equations is an integrable discretization of the Krichever-Novikov equation. The dimension $n$ of the Lie point symmetry algebra satisfies $1 \le n \le 5$. The highest dimensions, namely $n=5$ and $n=4$ occur only in the integrable cases.