Coupled nonlinear Schrödinger equations for Langmuir and elecromagnetic waves and extension of their modulational instability domain

Abstract

A pair of coupled nonlinear Schrödinger equations for transverse and longitudinal waves has been derived. The coupling resulting from equalization of group velocities drives the Langmuir waves modulationally unstable for wavelengths shorter than ( m i /m e )½ λ D and also extends the domain of modulational instability of electromagnetic waves when relativistic effects are taken into account. Instability is found to occur also for the perturbation wavenumber domain in which both the uncoupled Langmuir and electromagnetic waves are modulationally stable. This is shown to be caused by resonant four-wave interaction l + t → l′ + t′ . The growth rate of the instability is, in general, of the order of but increases to the extent of a factor ( c/v th ) 2 near the resonance Solitary wave solutions are given. Depending on the relative values of the self-modulation and coupling coefficients, the Langmuir or the transverse or both the waves may be localized in space.