基于对称性的非局部相关PDE系统的构造方法
A Symmetry-Based Method for Constructing Nonlocally Related PDE Systems
摘要：
非局部相关的偏微分方程（PDE）系统在分析给定的PDE系统中是有用的。已知一个偏微分方程系统的每一个局部守恒定律系统地产生一个非局部相关系统。本文介绍了一种新的构造非局部关联系统的补充方法。特别是，它示出了一个给定的偏微分方程系统的每个点对称系统地产生一个非局部相关的系统。例子包括非线性扩散方程，非线性波方程和非线性反应扩散方程的应用。因此，以前未知的非局部对称性表现为两个例子的非线性波方程。此外，由于所考虑的非线性反应扩散方程没有局部守恒律，以前的方法不产生非局部相关的系统，这样的方程。
英文摘要：
Nonlocally related partial differential equation (PDE) systems are useful in the analysis of a given PDE system. It is known that each local conservation law of a given PDE system systematically yields a nonlocally related system. In this paper, a new and complementary method for constructing nonlocally related systems is introduced. In particular, it is shown that each point symmetry of a given PDE system systematically yields a nonlocally related system. Examples include applications to nonlinear diffusion equations, nonlinear wave equations and nonlinear reaction-diffusion equations. As a consequence, previously unknown nonlocal symmetries are exhibited for two examples of nonlinear wave equations. Moreover, since the considered nonlinear reaction-diffusion equations have no local conservation laws, previous methods do not yield nonlocally related systems for such equations.
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