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

Author:

R. N. Garifullin

Keyword:

Nonlinear Sciences, Exactly Solvable and Integrable Systems, Exactly Solvable and Integrable Systems (nlin.SI)

journal:

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date:

2023-11-30 00:00:00

Abstract

The work deals with the qualification of semidiscrete hyperbolic type equations. We study a class of equations of the form $$\frac{du_{n+1}}{dx}=f\left(\frac{du_{n}}{dx},u_{n+1},u_{n}\right),$$ here the unknown function $u_n(x)$ depends on one discrete $n$ and one continuous $x$ variables. Qualification is based on the requirement of the existence of higher symmetries. The case is considered when the symmetry is of order 5 in continuous directions. As a result, a list of four equations with the required conditions is obtained. For one of the found equations, a Lax representation is constructed.