Asymptotic stability of small solitary waves for nonlinear Schrödinger equations with electromagnetic potential in R3

We consider the nonlinear magnetic Schrödinger equation for u:R3×R→C,iut=(i∇+A)2u+Vu+g(u),u(x,0)=u0(x), where A:R3→R3 is the magnetic potential, V:R3→R is the electric potential, and g=±|u|2u is the nonlinear term. We show that under suitable assumptions on the electric and magnetic potentials, if the initial data is small enough in H1, then the solution of the above equation decomposes uniquely into a standing wave part, which converges as t→∞ and a dispersive part, which scatters.