Nonlinear Dispersive Instabilities in Kelvin-Helmholtz Magnetohydrodynamic Flows

In this paper a weakly nonlinear theory of wave propagation in superposed fluids in the presence of magnetic fields is presented. The equations governing the evolution of the amplitude of the progressive waves are reported. The nonlinear evolution of Kelvin-Helmholtz instability (KHI) is examined in 2 + 1 dimensions in the context of magnetohydrodynamics (MHD). We study the envelope properties of the 2 + 1 dimensional wave packet. We converted the resulting nonlinear equation for the evolution of the wave packets in a 2 + 1 dimensional nonlinear Schroedinger (NLS) equation by using the function transformation method into a sine-Gordon equation, which depends only on one function, ζ. We obtained rather general classes of solutions of the equation in ζ which leads to rather general soliton solutions of the 2 + 1 dimensional NLS equation. This result contains interesting specific solutions such as N multiple solitons, propagational breathers and quadratic solitons