Periodic structures described by the perturbed Burgers-Korteweg-de Vries equation

TL;DR

This paper analyzes the perturbed Burgers-Korteweg-de Vries equation, revealing its non-integrability and constructing symmetry reductions, while discovering new periodic analytical structures relevant to nonlinear wave phenomena.

Contribution

It provides a symmetry analysis and introduces new periodic solutions for the perturbed Burgers-Korteweg-de Vries equation, expanding understanding of its nonlinear wave solutions.

Findings

The equation is non-integrable according to Painlevé analysis.

Constructed classical and nonclassical symmetries and reductions.

Discovered new types of periodic analytical structures.

Abstract

We study the perturbed Burgers-Korteweg-de Vries equation. This equation can be used for the description of nonlinear waves in a liquid with gas bubbles and for the description of nonlinear waves on a fluid layer flowing down an inclined plane. We investigate the integrability of this equation using the Painlev\'{e} approach. We show that the perturbed Burgers-Korteweg-de Vries equation does not belong to the class of integrable equations. Classical and nonclassical symmetries admitted by this equation and corresponding symmetry reductions are constructed. New types of periodic analytical structures described by the Burgers-Korteweg-de Vries equation are found.