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

We classify semidiscrete equations of hyperbolic type. We study the class of equations of the form du(n+1)/dx = f(du(n)/dx , u(n+1,) u(n)) where the unknown function u(n)(x) depends on one discrete (n) and one continuous ( x) variables. The classification is based on the requirement that generalized symmetries exist in the discrete and continuous directions. We consider the case where the symmetries are of order 3 in both directions. As a result, a list of equations with the required conditions is obtained.