Abstract

We classify semidiscrete equations of hyperbolic type. 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 (\(n\)) 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.

中文翻译：

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双曲型半离散方程的分类。三阶对称性的情况

摘要

我们对双曲型半离散方程进行分类。我们研究以下形式的方程组

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

其中未知函数\(u_n(x)\)取决于一个离散变量 ( \(n\) ) 和一个连续变量( \(x\) )。该分类基于离散和连续方向上存在广义对称性的要求。我们考虑对称性在两个方向上都是3 阶的情况。结果，获得了具有所需条件的方程列表。