Statistical description of the motion of particles trapped by a nonlinear wave

In the framework of the Hamilton formalism, a nonlinear theory is developed for the self-consistent motion of particles trapped in the potential wells of a nonlinear periodic wave (“pencil-box” model). The effective potential of the binary nonlinear interaction of the particles is constructed and used to derive a kinetic equation of Fokker–Planck type. A study is made of the kinetics of the trapped particles in the ergodic layer near the separatrix and its influence on the general kinetics of all the trapped particles. The time of relaxation of the trapped particles to an equilibrium distribution is found.