Nonlinear dispersion and adiabatic dynamics of waves with trapped particles

A Lagrangian of a general nondissipative quasiperiodic wave in collisionless plasma is obtained in a compact form with a transparent physical meaning. Langmuir waves with trapped electrons are studied as a paradigmatic example. The general nonlinear dispersion is derived, yielding various kinetic nonlinearities as particular cases; also, the action conservation theorem is restated. In the case of deeply trapped electrons, different regimes are realized depending on the parameter $S$, which is the ratio of the energy flux carried by trapped particles to that carried by passing particles. At small $S$ the waves are stable and exhibit group velocity splitting. At large $S$ the trapped-particle modulational instability (TPMI) develops, in contrast with the existing theories of the TPMI yet in agreement with the general sideband instability theory. Remarkably, these effects are not captured by the nonlinear Schr\"odinger equation, which is traditionally considered as a universal model of wave self-action.