Approximate symmetry reduction approach: infinite series reductions to the KdV-Burgers equation

TL;DR

This paper develops an approximate symmetry reduction method to derive infinite series solutions for the (1+1)-dimensional KdV-Burgers equation under weak dispersion and dissipation, linking solutions to special functions and Painlevé equations.

Contribution

It introduces a series reduction approach for the KdV-Burgers equation, enabling systematic derivation of solutions involving special functions and Painlevé equations.

Findings

Zero-order solutions relate to Painlevé and elliptic functions.

Higher-order solutions are obtained via linear ODEs.

Series reduction provides a structured way to approximate solutions.

Abstract

For weak dispersion and weak dissipation cases, the (1+1)-dimensional KdV-Burgers equation is investigated in terms of approximate symmetry reduction approach. The formal coherence of similarity reduction solutions and similarity reduction equations of different orders enables series reduction solutions. For weak dissipation case, zero-order similarity solutions satisfy the Painlev\'e II, Painlev\'e I and Jacobi elliptic function equations. For weak dispersion case, zero-order similarity solutions are in the form of Kummer, Airy and hyperbolic tangent functions. Higher order similarity solutions can be obtained by solving linear ordinary differential equations.