Asymptotic Stability and Completeness in the Energy Space for Nonlinear Schrödinger Equations with Small Solitary Waves

In this paper, we study a class of nonlinear Schrödinger equations (NLS) which admit families of small solitary wave solutions. We consider solutions which are small in the energy space H, and decompose them into solitary wave and dispersive wave components. The goal is to establish the asymptotic stability of the solitary wave and the asymptotic completeness of the dispersive wave. That is, we show that as t → ∞, the solitary wave component converges to a fixed solitary wave, and the dispersive component converges strongly inH to a solution of the free Schrödinger equation. We briefly supply some background. Solutions of dispersive partial differential equations (with repulsive nonlinearities) tend to spread out in space, although they often have conserved L mass. There has been extensive study of this phenomenon, usually referred to as scattering theory. These equations include Schrödinger equations, wave equations, and KdV equations. However, such equations can also possess solitary wave solutions which have localized spatial profiles that are constant in time (e.g., if the nonlinearity is attractive or if a linear potential is present). To understand the asymptotic dynamics of general solutions, it is essential to study the interaction between the solitary waves and the dispersive waves. The matter becomes more involved when the linearized operator around the solitary wave possesses multiple eigenvalues, which correspond to excited states. The interaction between eigenstates (mediated by the nonlinearity) is very delicate, and few results are known.