  The evolution of adiabatic waves with autoresonant trapped particles is
described within the Lagrangian model developed in Paper I, under the
assumption that the action distribution of these particles is conserved, and,
in particular, that their number within each wavelength is a fixed independent
parameter of the problem. One-dimensional nonlinear Langmuir waves with deeply
trapped electrons are addressed as a paradigmatic example. For a stationary
wave, tunneling into overcritical plasma is explained from the standpoint of
the action conservation theorem. For a nonstationary wave, qualitatively
different regimes are realized depending on the initial parameter $S$, which is
the ratio of the energy flux carried by trapped particles to that carried by
passing particles. At $S < 1/2$, a wave is stable and exhibits group velocity
splitting. At $S > 1/2$, 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 but misses the
trapped-particle oscillation-center inertia.
