Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schrodinger equation

TL;DR

This paper proves the local uniqueness and orbital stability of solitary waves for the one-dimensional mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger equation by reformulating the variational problem in the non-relativistic limit.

Contribution

It improves the understanding of the stability and uniqueness of solitary waves by reformulating the variational problem and analyzing the non-relativistic limit.

Findings

Established orbital stability of solitary waves.

Proved local uniqueness of the solutions.

Reformulated the variational problem in the non-relativistic regime.

Abstract

For the one-dimensional mass-critical/supercritical pseudo-relativistic nonlinear Schrodinger equation, a stationary solution can be constructed as an energy minimizer under an additional kinetic energy constraint and the set of energy minimizers is orbitally stable in \cite{BGV}. In this study, we proved the local uniqueness and established the orbital stability of the solitary wave by improving that of the energy minimizer set. A key aspect thereof is the reformulation of the variational problem in the non-relativistic regime, which we consider to be more natural because the proof extensively relies on the subcritical nature of the limiting model. Thus, the role of the additional constraint is clarified, a more suitable Gagliardo-Nirenberg inequality is introduced, and the non-relativistic limit is proved. Subsequently, this limit is employed to derive the local uniqueness and orbital stability.