Relaxation of Solitons in Nonlinear Schrodinger Equations with Potential

TL;DR

This paper proves the asymptotic stability of trapped solitons in nonlinear Schrödinger equations with external potentials, showing they relax to equilibrium with dynamics similar to damped Newtonian motion.

Contribution

It establishes conditions under which trapped solitons are asymptotically stable and derives effective equations of motion including dissipation.

Findings

Trapped solitons are asymptotically stable under certain conditions.

Solutions near a soliton resemble a moving soliton relaxing to equilibrium.

The soliton's motion follows a damped Newtonian-like equation.

Abstract

In this paper we study dynamics of solitons in the generalized nonlinear Schr\"odinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to $\mathcal{L}^2-$unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.