Axiomatic approach to wave-particle interactions and its applications to waves with trapped particles

A general axiomatic approach is developed that yields ponderomotive Lagrangians for wave-particle collisionless interactions deductively and, often, without even referring to the Maxwell-Vlasov system [PRL {\bf 107} (2011) 035005; Phys. Plasmas {\bf 12} (2012) 012102, 102103, 102104]. From those, nonlinear dispersion relations and dynamic equations are derived, and, as a spin-off, the long-standing controversies are resolved pertaining to photon properties in dielectric medium. Langmuir waves with trapped electrons are studied as a paradigmatic example. For the case of deeply trapped electrons in particular, action conservation predicts different regimes depending on the energy flux $S$ carried by trapped particles. For example, the trapped-particle modulational instability (TPMI) can develop just due to large $S$, in contrast with the existing theories. Remarkably, this effect is not captured by the nonlinear Schr\"odinger equation, which is traditionally considered as a universal model of wave self-action.