Modulations of slow sausage surface waves travelling along a magnetized slab

Abstract

In this paper we consider a set of nonlinear MHD equations that admits in a linear approximation a solution in the form of a slow sausage surface wave travelling along an isolated magnetic slab. For a wave of small but finite amplitude, we investigate how a slowly varying amplitude is modulated by nonlinear self-interactions. A stretching transformation shows that, at the lowest order of an asymptotic expansion, the original set of equations with appropriate boundary conditions (free interfaces) can be reduced to the cubic nonlinear Schrödinger equation, which determines the amplitude modulation. We study analytically and numerically the evolution of impulsively generated waves, showing a transition of the initial states into a train of solitons and periodic waves. The possibility of the existence of solitary waves in the solar atmosphere is also briefly discussed.