Weakly nonlinear theory of coherent Langmuir waves. II. Wave particle interaction

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Abstract
A weakly nonlinear analysis was carried out in Part 1 in the case of Langmuir waves, by neglecting the tail of the distribution function where particle velocities can be equal to phase velocities of waves. The treatment is extended in Part 2 by including particles of all velocities, so that resonant wave particle interaction becomes possible. The asymptotic theory of Krylov and Bogolioubov is applied to the Vlasov system of equations after writing it in terms of unknown functions which are constants of the motion of the linearized equations. A system of such constants can be found with the help of a suitable expansion in normal modes, which differs from Van Kampen’s expansion by the fact that weakly damped waves are treated on the same footing as weakly growing waves. It is then found that the free streaming portions of the distribution functions can no longer be left out, and the final result is a system of equations on both the wave amplitudes and the free streaming perturbations, which are coupled through resonant wave particle interaction.