This paper belongs to a group of work in the intersection of symbolic computation and group analysis aiming for the symbolic analysis of differential equations. The goal is to extract important properties without finding the explicit general solution. In this contribution, we introduce the algorithmic verification of nonlinear superposition properties and its implementation. More exactly, for a system of nonlinear ordinary differential equations of first order with a polynomial right-hand side, we check if the differential system admits a general solution by means of a superposition rule and a certain number of particular solutions. It is based on the theory of Newton polytopes and associated symbolic computation. The developed method provides the basis for the identification of nonlinear superpositions within a given system and for the construction of numerical methods which preserve important algebraic properties at the numerical level.

翻译：本文属于符号计算与群分析交叉领域的一系列工作，旨在实现微分方程的符号分析，即在无需显式通解的情况下提取其重要性质。本研究引入了非线性叠加性质的算法验证及其实现方法。具体而言，对于具有多项式右端项的非线性一阶常微分方程组，我们检验该微分系统是否通过叠加规则及若干特解形式存在通解。该方法基于牛顿多面体理论及其相关符号计算技术。所提出的方法为识别给定系统内的非线性叠加特性提供了基础，并为构建在数值层面保持重要代数性质的数值方法奠定了理论框架。