Relaxation of the Bose-condensate oscillations in the mesoscopic system at T=0

The general system is given of nonlinear equations describing dissipationless evolution of the oscillating Bose-condensate. The relaxation of transverse oscillations of the condensate in a trap of the cylindric symmetry is considered. The evolution occurs due to parametric resonance coupling the transverse mode with the longitudinal ones. The nonlinear rescattering in the subsystem of discrete longitudinal modes results in suppression of the return of energy, yielding dissipationless nonmonotonic relaxation of transverse oscillations in the condensate.