Eduard Kirr, Arghir Zarnescu
On the asymptotic stability of bound states in 2D cubic Schroedinger equation
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ABSTRACT. We consider the cubic nonlinear Schr\"{o}dinger equation in two space
dimensions with an attractive potential. We study the asymptotic
stability of the nonlinear bound states, i.e. periodic in time
localized in space solutions. Our result shows that all solutions
with small, localized in space initial data, converge to the set of
bound states. Therefore, the center manifold in this problem is a
global attractor. The proof hinges on dispersive estimates that we
obtain for the non-autonomous, non-Hamiltonian, linearized dynamics
around the bound states.