We consider nonlinear Schrödinger equationswhere d ≥ 3 and β is smooth. We prove that symmetric finite energy solutions close to orbitally stable ground states converge to a sum of a ground state and a dispersive wave as t → ∞ assuming the so called Fermi Golden Rule (FGR) hypothesis. We improve the "sign condition"required in a recent paper by Gang Zhou and I.M.Sigal.

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We establish an asymptotic stability result for Toda lattice soliton solutions, by making use of a linearized Bäcklund transformation whose domain has codimension one. Combining a linear stability result with a general theory of nonlinear stability by Friesecke and Pego for solitary waves in lattice equations, we conclude that all solitons in the Toda lattice are asymptotically stable in an exponentially weighted norm. In addition, we determine the complete spectrum of an operator naturally associated with the Floquet theory for these lattice solitons.

We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schrödinger equationsin the energy class. This problem was studied by in the 3-dimensional case using the endpoint Strichartz estimate.To prove asymptotic stability of solitary waves, we need to show that a dispersive part v(t, x) of a solution belongs to L 2 t (0, ∞; X) for some space X. In the 1-dimensional case, this property does not follow from the Strichartz estimate alone.In this paper, we prove that a local smoothing estimate of Kato type holds globally in time and combine the estimate with the Strichartz estimate to show (1+x 2 ) −3/4 v L ∞ x L 2 t < ∞, which implies the asymptotic stability of a solitary wave.

We prove the nonlinear stability of the KdV solitary waves considered as solutions of the KP-II equation, with respect to periodic transverse perturbations. Our proof uses a Miura transform which sends the solutions of an mKP-II equation to solutions of the KP-II equation.