Framework for potential systems and nonlocal symmetries: Algorithmic approach

Abstract

An algorithmic framework is presented to find an extended tree of nonlocally related systems for a given system of differential equations (DEs). Each system in an extended tree is equivalent in the sense that the solution set for any system in a tree can be found from the solution set for any other system in the tree. Useful conservation laws play an essential role in the construction of an extended tree. A useful conservation law yields potential variables and equivalent nonlocally related potential systems and subsystems for any given system. Nonlocal symmetries for a given system of DEs can arise from any system in its extended tree. We construct extended trees for the systems of planar gas dynamics and nonlinear telegraph equations, and in both cases obtain new nonlocal symmetries. More importantly, due to the equivalence of solution sets, any coordinate-independent method of analysis (qualitative, numerical, perturbation, etc.) can be applied to any system within the tree, and may yield simpler computations and/or results that cannot be obtained when the method is directly applied to the given system.