The extended research for constructing three-dimensional nonlocally related partial differential equation systems

Abstract

The coordinate-independent analysis (qualitative, numerical, perturbation, etc.) on a given partial differential equation (PDE) may be effectively facilitated if its nonlocally related PDE systems can be constructed. The present research extendes their construction in three-dimensional situations, including the reduced potential system (RPS) based on the divergence-type conservation law (CL) and the inverse potential system (IPS) based on the differential invariant (DI). For the RPS, the breakthrough is to prove that the potential system with the algebraic gauge constraint can further reduce the potential without weakening the solution space. The RPS with a more compact structure not only simplifies the computation of symmetric properties (equivalent transformations, nonlocal symmetries), but also promotes to produce more valuable results (nonlocal CLs and the expansion of the nonlocal CLs based-RPS). For the IPS, the more critical breakthrough is to generalize the previous theorems from two-dimensional to three-dimensional. It consists of nonlocal symmetries’ identification by the IPS, and the IPS’s expansion by the solvable Lie algebra. As the application, this research is demonstrated with a typical three-dimensional PDE with the power law nonlinearity and multiple parameters. Once the parameters are determined, our analysis corresponds to several equations describing turbulence, shock waves, and weakly diffracted waves. Concretely, its RPSs, IPSs, their expansions and associatied symmetric properties are derived. Consequently, these generated systems are nonlocally related to the demonstrated equation, constituting an expanded tree of nonlocally related PDE systems. Moreover, this research can aslo be applied to other three-dimensional PDEs that admit CLs or DIs.