Nonlinear Schr\"odinger equation on a unit ball in one and two dimensions

TL;DR

This paper studies the nonlinear Schr"odinger equation on a unit ball in 1D and 2D, analyzing stability, bifurcations, and the absence of scattering, revealing how solutions behave under various perturbations.

Contribution

It provides a detailed analysis of ground state stability, bifurcation behavior, and soliton resolution for the nonlinear Schr"odinger equation on a bounded domain.

Findings

Ground states are stable in subcritical and critical cases.

Supercritical ground states split into stable and unstable branches.

Solutions do not exhibit scattering or radiation, confirming soliton resolution.

Abstract

We consider the nonlinear Schr\"odinger equation on a unit ball in one and two dimensions with Dirichlet boundary conditions, which have stabilizing effect on solutions behavior. In particular, we confirm that the ground state solutions are stable in subcritical and critical cases, and in the supercritical case the ground state solutions split into a stable and an unstable branch. Perturbations of a ground state on the stable branch keep solutions near a corresponding ground state with very small oscillation around it, while perturbations of the unstable branch make solutions either blow up in finite time, if perturbations have an amplitude large than the height of the ground state, or oscillate between two states, if perturbations have an amplitude smaller than the original ground state. We also observe that this equation does not have any scattering or radiation, and thus, the soliton resolution holds for all data, splitting solutions into coherent structures such as ground state solutions even for very small initial data.