Soliton dynamics for the nonlinear Schr\"odinger equation with magnetic field

TL;DR

This paper investigates the behavior of soliton solutions to the nonlinear Schrödinger equation under magnetic and electric fields, analyzing their concentration along trajectories governed by Newtonian dynamics with electromagnetic forces.

Contribution

It introduces a novel analysis of soliton dynamics in the presence of magnetic fields, linking PDE solutions to classical Newtonian equations with electromagnetic forces.

Findings

Solutions concentrate along trajectories governed by Newton's laws with electromagnetic forces.

The study extends understanding of soliton behavior in magnetic fields.

Provides a mathematical framework connecting quantum PDEs and classical mechanics.

Abstract

The semiclassical limit of a nonlinear focusing Schr\"odinger equation in presence of nonconstant electric and magnetic potentials V,A is studied by taking as initial datum the ground state solution of an associated autonomous elliptic equation. The concentration curve of the solutions is a parameterization of the solutions of a Newton ODE involving the electric force as well as the magnetic force via the Lorenz law of electrodynamics.