1 research outputs found
Generalization of nonlocally related partial differential equation systems: unknown symmetric properties and analytical solutions
Symmetry, which describes invariance, is an eternal concern in mathematics
and physics, especially in the investigation of solutions to the partial
differential equation (PDE). A PDE's nonlocally related PDE systems provide
excellent approaches to search for various symmetries that expand the range of
its known solutions. They composed of potential systems based on conservation
laws and inverse potential systems (IPS) based on differential invariants. Our
study is devoted to generalizing their construction and application in
three-dimensional circumstances. Concretely, the potential of the algebraic
gauge-constrained potential system is simplified without weakening its solution
space. The potential system is extended via nonlocal conservation laws and
double reductions. Afterwards, nonlocal symmetries are identified in the IPS.\@
The IPS is extended by the solvable Lie algebra and type \Rmnum{2} hidden
symmetries. Besides, systems among equations can be connected via Cole-Hopf
transformation.\@ Ultimately, established and extended systems embody rich
symmetric properties and unprecedented analytical solutions, and may even
further facilitate general coordinate-independent analysis in qualitative,
numerical, perturbation, etc., this can be illustrated by several Burgers-type
equations