Reduction operators of Burgers equation

TL;DR

This paper systematically analyzes reduction operators and nonclassical reductions of the Burgers equation, providing new proofs, complete classifications, and linking Lie reductions to heat equation solutions.

Contribution

It offers a comprehensive classification of nonclassical reductions of the Burgers equation and introduces a new proof for the 'no-go' case of regular reduction operators.

Findings

Complete description of nonclassical reductions to ODEs.

Representation of reduction operator coefficients via solutions of the Burgers equation.

All Lie reductions are equivalent to heat equation reductions via Hopf-Cole transformation.

Abstract

The solution of the problem on reduction operators and nonclassical reductions of the Burgers equation is systematically treated and completed. A new proof of the theorem on the special "no-go" case of regular reduction operators is presented, and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case. All possible nonclassical reductions of the Burgers equation to single ordinary differential equations are exhaustively described. Any Lie reduction of the Burgers equation proves to be equivalent via the Hopf-Cole transformation to a parameterized family of Lie reductions of the linear heat equation.