A systemic method to construct the high order nonlocal symmetries

We propose a systemic method of applying the auxiliary systems of original equations to find the high order nonlocal symmetries of nonlinear evolution equation. In order to validate the effectiveness of the method, some examples are presented.

💡 Research Summary

The paper introduces a systematic procedure for constructing high‑order nonlocal symmetries of nonlinear evolution equations by exploiting auxiliary systems associated with the original equations. Traditional symmetry analysis, based on Lie groups, is limited to local point or contact symmetries, and while first‑order nonlocal symmetries (often obtained by introducing a single potential variable) have been studied, a general method for higher‑order nonlocal symmetries has been lacking. The authors fill this gap by embedding the original PDE together with an auxiliary system—typically a potential equation, a conservation law, or a transformation relation—into an enlarged system that includes new dependent variables and, crucially, their higher‑order derivatives.

The methodology proceeds in four main steps. (1) Auxiliary System Construction: For a given evolution equation (E(x,u^{(n)})=0), one selects an auxiliary relation (A(x,u^{(m)},\psi^{(k)})=0) that links a new variable (\psi) (or a set of such variables) to the original fields. The auxiliary system must be invertible (so that (\psi) can be expressed in terms of (u) and its derivatives) and closed under differentiation. (2) Extended Symmetry Analysis: The combined system (\mathcal{S}={E,A}) is treated as a standard PDE system. One writes a general prolonged symmetry generator \