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
$$ \frac{du_{n+1}}{dx}=F\biggl(\frac{du_{n}}{dx},u_{n+1},u_{n}\biggr), $$
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.