Analytical and numerical studying of the perturbed Korteweg--de Vries equation

TL;DR

This paper investigates the perturbed Korteweg--de Vries equation's integrability, symmetries, exact solutions, and stability, providing new insights into its mathematical structure and physical applications in nonlinear wave phenomena.

Contribution

It establishes conditions for integrability, finds new symmetries, constructs exact solutions, and analyzes their stability, advancing understanding of the perturbed KdV equation.

Findings

Conditions for integrability are identified.

New classical and nonclassical symmetries are discovered.

Exact solutions expressed via trigonometric and Airy functions are constructed.

Abstract

The perturbed Korteweg--de Vries equation is considered. This equation is used for the description of one--dimensional viscous gas dynamics, nonlinear waves in a liquid with gas bubbles and nonlinear acoustic waves. The integrability of this equation is investigated using the Painlev\'e approach. The condition for parameters for the integrability of the perturbed Korteweg--de Vries equation equation is established. New classical and nonclassical symmetries admitted by this equation are found. All corresponding symmetry reductions are obtained. New exact solutions of these reductions are constructed. They are expressed via trigonometric and Airy functions. Stability of the exact solutions of the perturbed Korteweg--de Vries equation is investigated numerically.