Extended symmetry analysis of two-dimensional degenerate Burgers equation

Abstract

We carry out the extended symmetry analysis of a two-dimensional degenerate Burgers equation. Its complete point-symmetry group is found using the algebraic method, and all its generalized symmetries are proved equivalent to its Lie symmetries. We also prove that the space of conservation laws of this equation is infinite-dimensional and is naturally isomorphic to the solution space of the (1+1)-dimensional backward linear heat equation. Lie reductions of the two-dimensional degenerate Burgers equation are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We additionally consider solutions that also satisfy an analogous nondegenerate Burgers equation. In total, we construct four families of solutions of two-dimensional degenerate Burgers equation that are expressed in terms of arbitrary (nonzero) solutions of the (1+1)-dimensional linear heat equation. Various kinds of hidden symmetries and hidden conservation laws (local and potential ones) are discussed as well. As a by-product, we exhaustively describe generalized symmetries, cosymmetries and conservation laws of the transport equation, also called the inviscid Burgers equation, and construct new invariant solutions of the nonlinear diffusion and diffusion–convection equations with power nonlinearities of degree −1/2.