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

Abstract

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.

34 pages

References in corpus (2)

• On the asymptotic stability of bound states in 2D cubic Schroedinger equation
• Asymptotic stability of small solitons to 1D NLS with potential