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We consider a class of nonlinear Schr\"odinger equation in two spacedimensions with an attractive potential. The nonlinearity is local but rathergeneral encompassing for the first time both subcritical and supercritical (in$L^2$) nonlinearities. We study the asymptotic stability of the nonlinear boundstates, i.e. periodic in time localized in space solutions. Our result showsthat all solutions with small initial data, converge to a nonlinear boundstate. Therefore, the nonlinear bound states are asymptotically stable. Theproof hinges on dispersive estimates that we obtain for the time dependent,Hamiltonian, linearized dynamics around a careful chosen one parameter familyof bound states that "shadows" the nonlinear evolution of the system. Due tothe generality of the methods we develop we expect them to extend to the caseof perturbations of large bound states and to other nonlinear dispersive wavetype equations.

Asymptotic stability of ground states in 2D nonlinear Schr\"odinger equation including subcritical cases