On the nonlinear development of the Langmuit modulational instability

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Abstract
We consider the nonlinear development of a long-wavelength finite-amplitude Langmuir wave. The wavenumber k0 of the initial Langmuir wave is chosen such that the three-wave decay is forbidden. We then describe the coupling of the initial Langmuir wave to Stokes and anti-Stokes Langmuir perturbations (with wavenumbers k0 ∓ ks) due to the presence of a low-frequency density perturbation of wavenumber ks. We then show that for a wide range of experimental conditions, the Stokes and anti-Stokes Langmuir waves are generated with wavenumbers well separated from k0. In order to describe the nonlinear evolution of these perturbations and the pump wave we make the static approximation for the ions and describe the high-frequency waves by three distinct wave envelopes. These coupled nonlinear differential equations are then solved exactly for a number of special cases. For the temporal evolution, we obtain periodic solutions and, when damping is included, we find a slow exponential decay of the amplitudes with a corresponding increase in the nonlinear period of oscillation. The stationary spatially varying solutions are shown to include four basic types of behaviour: periodic, solitary wave, phase jump and shock-like profiles. These latter solutions are of interest since they are obtained for zero dissipation and for a coherent wave interaction.