# Extended symmetry analysis of two-dimensional degenerate Burgers equation

## TL;DR

This paper performs a comprehensive symmetry analysis of a two-dimensional degenerate Burgers equation, revealing its symmetry structure, conservation laws, and constructing new solutions using Lie reduction methods.

## Contribution

It provides the complete point-symmetry group, classifies all generalized symmetries, and constructs explicit solutions related to the heat equation, advancing understanding of this degenerate PDE.

## Key findings

- Infinite-dimensional space of conservation laws is isomorphic to the heat equation solution space.
- Four families of solutions expressed via heat equation solutions are constructed.
- Hidden symmetries and conservation laws are identified and discussed.

## 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.