The modulational instability of Langmuir waves in unmagnetised plasmas is reviewed for the cases when a pump consist of two monochromatic or a large number of random modes. It is demonstrated that the correct theory for the modulational instability operates with `renormalised' equations for the linear dielectric function as well as for the effective third--order plasma response. This renormalisation is due to so--called interference terms. The appearance of interference terms is a specific feature of the multi--mode modulational instability in comparison with the well known instability of a single mode. All calculations use a simple and universal formalism including new methods developed for description of the modulational effects in arbitrary media. The modulational instability of two pump Langmuir modes is considered for the case of comparatively small instability rates, when `renormalised' expressions for linear and nonlinear plasma polarisation responses provide the maximum effect on the instability development. For instabilities of the broad spectra of random waves, the integral equations are presented for perturbations of wave field correlation functions. In the description of the modulational instability of random wave packets these equations play the same role as the set of coupled equations for the fields of modulational perturbations in the case of two monochromatic pumps. Rates and thresholds of the instabilities are found in various limits.

1. Introduction.........................................................................................................................376

2. General nonlinear formalism...................................................................................................380

3. Interaction of two monochromatic pumps...................................................................................383

3.1. Interference terms........................................................................................................................383
3.2. Steady-state solutions...............................................................................................................385
3.3. Modulational matrix.....................................................................................................................388

4. Two monochromatic pumps with large frequency gap...................................................................389

4.1. Frequency analysis.....................................................................................................................389
4.2. Basic expressions........................................................................................................................391
4.3. Zeroth approximation..................................................................................................................393
4.4. Equations for modulational perturbations..................................................................................394

5. Rates of two-pump instability of Langmuir waves.......................................................................396

5.1. Nonlinear frequency shift.............................................................................................................396
5.2. Modulational perturbations.........................................................................................................398
5.3. Instability rates............................................................................................................................401

6. Interaction of broad wave packets..........................................................................................404

6.1. General equations........................................................................................................................404
6.2. Regular and random fields...........................................................................................................406
6.3. Perturbations of correlation functions.........................................................................................409

7. Rates of instability of broad wave packets................................................................................412

7.1. One-dimensional equations.......................................................................................................412
7.2. Rates in the one-dimensional case...........................................................................................415
7.3. Conditions of the one-dimensional instability..........................................................................418
7.4. Correlations functions in the isotropic case...............................................................................421
7.5. Instability rates in the isotropic case.........................................................................................422

8. Conclusions........................................................................................................................424
Acknowledgements.................................................................................................................427
References............................................................................................................................427

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