Long-time asymptotics of solutions of the third-order nonlinear evolution equation governing wave propagation in relaxing media

Abstract

A classical Cauchy problem for a third-order nonlinear evolution equation is considered. This equation describes the propagation of weakly nonlinear waves in relaxing media. The global existence and uniqueness of its solutions is proved and the solution is constructed in the form of a series in a small parameter present in the initial conditions. Its long-time asymptotics is calculated, which shows the presence of two solitary wave pulses traveling in opposite directions and diffusing in space. Each of them is governed by Burgers’ equation with a transfer.