Classification of semidiscrete hyperbolic type equations. The case of third order symmetries

Abstract

In this paper, a classification of semidiscrete equations of hyperbolic type is carried out. We study the class of equations of the form $$\frac{d{u}_{n+1}}{dx}=f\left(\frac{d{u}_n}{dx},u_{n+1},u_n\right),$$ here is the unknown function $u_n(x)$ depends on one discrete $n$ and one continuous $x$ variables. The classification is based on the requirement for the existence of higher symmetries in the discrete and continuous directions. The case is considered when the symmetries are of order 3 in both directions. As a result, a list of equations with the required conditions is obtained.

arXiv comment15 pages (in Russian)

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• Bäcklund transformation for the Krichever-Novikov equation
• Generalized symmetry classification of discrete equations of a class depending on twelve parameters
• Generalized symmetries and integrability conditions for hyperbolic type semi-discrete equations
• An unusual series of autonomous discrete integrable equations on the square lattice
• Non-Point Invertible Transformations and Integrability of Partial Difference Equations

Cited by in corpus (1)

• Classification of semidiscrete hyperbolic type equations. The case of fifth order symmetries