Enhanced symmetry analysis of two-dimensional Burgers system

TL;DR

This paper performs an enhanced symmetry analysis of the two-dimensional Burgers system, finding its symmetry group, constructing new solutions, and exploring conservation laws and hidden symmetries.

Contribution

It introduces an enhanced algebraic method to fully determine the symmetry group and constructs new solutions, including those related to the heat equation.

Findings

Complete point symmetry group identified

New Lie invariant solutions constructed

No local conservation laws, but hidden ones found

Abstract

We carry out enhanced symmetry analysis of a two-dimensional Burgers system. The complete point symmetry group of this system is found using an enhanced version of the algebraic method. Lie reductions of the Burgers system are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We prove that this system admits no local conservation laws and then study hidden conservation laws, including potential ones. Various kinds of hidden symmetries (continuous, discrete and potential ones) are considered for this system as well. We exhaustively describe the solution subsets of the Burgers system that are its common solutions with its inviscid counterpart and with the two-dimensional Navier-Stokes equations. Using the method of differential constraints, which is particularly efficient for the Burgers system, we construct a number of wide families of solutions of this system that are expressed in terms of solutions of the (1+1)-dimensional linear heat equation although they are not related to the well-known linearizable solution subset of the Burgers system.