Integrable Unstable Model for Interaction of Langmuir Waves with Acoustic Waves in Plasmas

Abstract

We study here the dynamical interaction of the electrostatic waves (ESW) with the ion acoustic waves (IAW), in one spatial dimension, by deriving from the hydrodynamic Poisson-Maxwell equations for the plasma an integrable model for which we solve an initial/boundary value problem. The boundary value problem consists in prescribing the incoming ESW at one end (say x = +∞) and requiring no incoming wave at the other end (x = −∞). The initial datum consists in prescribing the IAW at time zero.

An one-dimensional model of coupled nonlinear equations is derived to describe the interaction of the Langmuir waves with the acoustic waves through the ponderomotive force. This model includes nonlinearity and dispersion of the acoustic wave and accounts for a resonant scattering of the electrostatic wave. The integrability of the model allows us to understand the instability of the Langmuir wave as i) a mutual trapping of the acoustic wave and the scattered Langmuir wave, ii) an asymptotic time evolution of the solution towards the formation of a local singularity of the Langmuir wave (collapsing).