Evolution of long waves and chaos in magnetohydrodynamics

Abstract

The evolution equation for a wavepacket travelling on the surface of a conducting fluid of finite depth in 2+1 dimensions in the long-wave approximation is studied in the context of magnetohydrodynamics. The amplitude equation thus obtained is the well-known Kadomtsev–Petviashvili equation with modified coefficients in the presence of a tangential magnetic field. By taking the double limit of this equation, Schrödinger–Poisson type equations are obtained. A nonlinear evolution equation is sought, in order to study Rayleigh–Taylor instability. It is shown that the magnetic field and surface tension have a stabilizing influence on the formation of bubbles arising owing to this instability. By incorporating forcing and damping terms in the Kadomtsev–Petviashvili equation, a condition for the existence of transverse homoclinic orbits giving rise to chaotic motions is obtained by using the Melnikov function method. It is shown that the chaotic motions can be suppressed by applying a suitably strong magnetic field. A study of subharmonic bifurcation leading to surface waves is also undertaken. The tangential magnetic field has a stabilizing influence on such motions.