Framework for nonlocally related PDE systems and nonlocal symmetries: extension, simplification, and examples

Any PDE system can be effectively analyzed through consideration of its tree of nonlocally related systems. If a given PDE system has n local conservation laws, then each conservation law yields potential equations and a corresponding nonlocally related potential system. Moreover, from these n conservation laws, one can directly construct 2 − 1 independent nonlocally related systems by considering these potential systems individually (n singlets), in pairs (n(n − 1)/2 couplets),. . . , taken all together (one n-plet). In turn, any one of these 2 − 1 systems could lead to the discovery of new nonlocal symmetries and/or nonlocal conservation laws of the given PDE system. Moreover, such nonlocal conservation laws could yield further nonlocally related systems. A theorem is proved that simplifies this framework to find such extended trees by eliminating redundant systems. The planar gas dynamics equations and nonlinear telegraph equations are used as illustrative examples. Many new local and nonlocal conservation laws and nonlocal symmetries are found for these systems. In particular, our examples illustrate that a local symmetry of a k-plet is not always a local symmetry of its “completed” n-plet (k < n). A new analytical solution, arising as an invariant solution for a potential Lagrange system, is constructed for a generalized polytropic gas.