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

TL;DR

This paper proves the asymptotic stability of small solitary waves for a nonlinear magnetic Schr"odinger equation in three dimensions, showing solutions decompose into a stable standing wave and a dispersive scattering component.

Contribution

It establishes the stability and scattering behavior of small solitary waves in a magnetic Schr"odinger setting with electric and magnetic potentials.

Findings

Small initial data lead to solutions decomposing into a stable standing wave and a dispersive part.

The standing wave component converges as time approaches infinity.

The dispersive component scatters, resembling free evolution at large times.

Abstract

We consider the nonlinear magnetic Schr\"odinger 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.