Nonlinear interactions of two compressional hydromagnetic waves

Abstract

Nonlinear interactions of two azimuthally symmetric compressional hydromagnetic waves propagating in a cylindrical waveguide filled with cold magnetized plasma are investigated. Two cases are considered: the nonlinear interaction of two identical oppositely propagating compressional waves and the nonlinear interaction of two compressional waves propagating with equal group velocities. In the first case the second-order perturbation fields generated through self- and mutual interactions of the waves are calculated and their effect on the otherwise-formed simple linear standing-wave pattern is studied. The possibility of observing a resonant nonlinear interaction is shown. In the second case, in order to describe the nonlinear evolution of the wave amplitudes, two coupled nonlinear Schrödinger (NLS) equations are presented. When excited individually, both the waves are seen to be modulationally stable; but when excited simultaneously, a strong nonlinear wave-wave coupling comes into play, which makes the waves modulationally unstable. The corresponding growth rate of the instability is also calculated.

Resonant interaction of a magneto-hydrodynamic wave with its second harmonic in a planar plasma filled waveguide

“…Similarly, substituting the expansion (8) in the set of equations 2 we get from the lowest order equations the following solutions…”

“…Finite amplitude effects in the propagation of torsional MHD waves in a cold magnetized cylindrical plasma column have been studied by Jones and Cheetham [6] and references therein. Self-interaction and interaction of two compressional MHD waves in a cylindrical plasma column have been studied by Ghosh and Das [7,8]. Many works on the nonlinear behaviour of compressional MHD waves have been devoted to the study of various nonlinear damping processes such as parametric decay [9] and mode conversion [10].…”