Perturbation of a simple wave: from simulation to asymptotics

Abstract

We consider a problem on perturbation of a simple (travelling) wave at the example of a nonlinear partial differential equation that models domain wall dynamics in the weak ferromagnets. The main attention is focused on the case when, for fixed constants coefficients, there exist many exact solutions in the form of a simple wave. These solutions are determined by an ordinary differential equation with boundary conditions at infinity. The equation depends on the wave velocity as a parameter. Suitable solutions correspond to the phase trajectory connecting the equilibria. The main problem is that the wave velocity is not uniquely determined by the coefficients of the initial equations. For an equation with slowly varying coefficients, the asymptotics of the solution is constructed with respect to a small parameter. In the considered case, the well-known asymptotic construction turns out to be ambiguous due to the uncertainty of the perturbed wave velocity. For unique identification of the velocity, we propose an additional restriction on the structure of the asymptotic solution. This restriction is a stability of the wave front is formulated on the base of numerical simulation of the original equation.