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

We classify the semidiscrete hyperbolic-type equations. We study the class of equations of the form

where the unknown function \(u_n(x)\) depends on one discrete variable \(n\) and one continuous variable \(x\). The classification is based on the requirement of the existence of higher symmetries. We consider the case where the symmetry has the fifth order in the continuous direction. As a result, we obtain a list of four equations with the required conditions, for each of which the higher symmetry in the discrete direction is written. For one of the obtained equations, we construct a Lax representation.