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

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 Painlevé 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.

On the dynamics of the zeros of solutions of the Airy equation

“…As a last remark, we would like to underline that the nonhomogeneous Airy equation possesses several applications in mathematical physics [9]. For example, it has a strict relation with the second member of the Burger's hierarchy [6]:…”

“…The previous equation will be considered in the next sections (see in particular equation ( 46)). Some of the solutions of ( 4) have been considered in [6] for the description of liquids with gas bubbles.…”