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 Poincaré map are studied.wafer. 6 In the last case, atoms are reflected by the attracting tails, of the van der Waals attraction (quantum reflection). 7,8 Such reflection is efficient when the normal component of the wavenumber of the atoms is small or comparable to the effective depth of the attraction potential, roughly, the distance at which the potential becomes comparable to the kinetic energy of the atom. To reduce the normal component most atomic mirrors are blazed at the grazing incidence. At grazing incidence, the efficiency of the quantum reflection can be enhanced by a surface covered with ridges. [9][10][11][12] Recently quantum reflection of matter-waves from a solid surface has been the subject of considerable interest both from the viewpoints of basic physics and BEC applications. Specifically, matter-wave dynamics near the solid surface can be a very sensitive probe for the Casimir force. 13 Meantime, atom chips, where a BEC is stored and manipulated near the solid substrate, open up new perspectives for application. 14-16 Coherent acceleration of matter-wave packet falling under gravity and bouncing off a modulated magnetic mirror showed the possibility to realize the Fermi acceleration with matter-waves. 17 This work is aimed at investigation of the dynamics of a matter-wave soliton near the solid surface under the action of a linear potential, originating from the attractive force of gravity. The effect of a solid surface is modeled by a reflecting delta-potential barrier. In real experiments such a barrier can be created by means of a laser light far-off blue-detuned from atomic transitions. The underlying mathematical model is based on the one-dimensional Gross-Pitaevskii equation (GPE) for the BEC with a negative atomic scattering length, when the GPE supports self-localized solution, the so-called matter-wave soliton.The paper is organized as follows. In Sec. 2, we describe interactions between the matter-wave soliton and delta-potential barrier. In Sec. 3, the Krylov-Bogoliubov method applied to the equation of motion for the center-of-mass coordinate of the soliton. In Sec. 4, we consider the Poincaré map for nonlinear resonance. In concluding Sec. 5, we summarize our results. 1450198-2 Int. J. Mod. Phys. B 2014.28. Downloaded from www.worldscientific.com by UNIVERSITY OF CALIFORNIA @ DAVIS on 02/07/15. For personal use only.

Communicated by (xxxxxxxxxx)The dynamics of a two-soliton molecule bouncing on the reflecting atomic mirror under the effect of gravity has been studied by analytical and numerical methods. The analytical description is based on the variational approximation. In numerical simulations, we observe the resonance oscillations of the two-soliton's center-of-mass position and width, induced by modulated atomic mirror. Theoretical predictions are verified by numerical simulations of the nonlocal Gross-Pitaevskii equation (GPE) and qualitative agreement between them is found. Hamiltonian dynamic system for a dipolar Bose-Einstein condensates (BECs) has been studied.