AI 中文总结
本文提出嵌套导数准则，结合刘维尔定理与伽罗瓦理论的分部积分方法，研究一维时空下一类满足$\boldsymbol{O}_t u=\boldsymbol{O}_x u$的非线性偏微分方程的基本解，该类解或助力物理系统演化研究。
AI 中文摘要
本文研究了一种特定准则，以确定在一维时空$(x,t)$中满足$\boldsymbol{\textit{O}}_t u=\boldsymbol{\textit{O}}_x u$的一类非线性偏微分方程，是否能针对空间坐标$x$建立基本解，该过程采用嵌套导数方法。其中，嵌套导数是通常高阶偏导数的推广，在微分算子层面是拉盖尔（Laguerre）算子的推广。结合刘维尔（Liouville）定理与伽罗瓦理论（Galois theory）的分部积分方法，可确定嵌套导数方程的解是否能针对$x$建立基本解，这类解或有助于更好地研究时空下可能物理系统的演化。
英文摘要
In this article, we have studied a particular criterion to establish if a particular class of nonlinear PDEs $\mathcal{O}_t u=\mathcal{O}_x u $ in unidimensional space-time $(x,t)$ admits elementary solutions respect to the spatial coordinate $x$, through the nested derivative method. Where a nested derivative is a generalization of the usual partial derivative of a order higher than one and, in terms of differentials operators, is a generalization of the Laguerre operator. This method, integrated with Liouville's theorem and Galois theory and method by integrate by parts, establishes if solution of nested derivative equations admits elementary solutions respect with $x$. These solutions could help to study better evolutions of possible physical systems in space-time.
