### Abstract

We consider the nonlinear development of a long-wavelength finite-amplitude Langmuir wave. The wavenumber k

_{0}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 k_{0}∓ k_{s}) due to the presence of a low-frequency density perturbation of wavenumber k_{s}. 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 k_{0}. 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.Original language | English |
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Pages (from-to) | 51-69 |

Number of pages | 19 |

Journal | Journal of Plasma Physics |

Volume | 21 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 1979 |

### Keywords

- mathematical techniques
- Langmuir waves
- plasmas

## Cite this

Bingham, R., & Lashmore-Davies, C. N. (1979). On the nonlinear development of the Langmuir modulational instability.

*Journal of Plasma Physics*,*21*(1), 51-69. https://doi.org/10.1017/S0022377800021644