Asymptotic stability of small solitary waves for nonlinear Schrödinger equations with electromagnetic potential in $\mathbb{R}^3$

We consider the nonlinear magnetic Schrödinger equation for $u: \mathbb{R}^3 \times \mathbb{R} \to \mathbb{C}$, $iu_t = (i \nabla + A)^2 u + V u + g(u), u(x,0) = u_0(x),$ where $A :\mathbb{R}^3 \to \mathbb{R}^3$ is the magnetic potential, $V : \mathbb{R}^3 \to \mathbb{R}$ is the electric potential, and $g = \pm | u |^2 u$ is the nonlinear term. We show that under suitable assumptions on the electric and magnetic potentials, if the initial data is small enough in $H^1$, then the solution of the above equation decomposes uniquely into a standing wave part, which converges as $t \to \infty$ and a dispersive part, which scatters.

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