Regular and chaotic dynamics of a matter-wave soliton near the atomic mirror

The dynamics of the soliton in a self-attractive Bose-Einstein condensate under the gravity are investigated. First, we apply the inverse scattering method, which gives rise to equation of motion for the center-of-mass coordinate of the soliton. We analyze the amplitude-frequency characteristic for nonlinear resonance. Applying the Krylov-Bogoliubov method for the small parameters the dynamics of soliton on the phase plane are considered. Hamiltonian chaos under the action of the gravity on the Poincar'e map are studied.