A Symmetry-Based Method for Constructing Nonlocally Related PDE Systems

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.

💡 Research Summary

The paper introduces a novel, symmetry‑based framework for generating nonlocally related partial differential equation (PDE) systems, complementing the well‑established method that relies on local conservation laws. The authors begin by recalling that each local conservation law of a given PDE system can be used to introduce potential (or auxiliary) variables, thereby constructing a nonlocal system that is equivalent to the original one but enriched with additional variables and equations. While powerful, this approach fails when a PDE admits no conservation laws, leaving a substantial class of nonlinear equations without a systematic nonlocal extension.

To overcome this limitation, the authors prove that every point symmetry of a PDE system can be exploited in a similar fashion. Starting from a point symmetry generator
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