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

Younghun Hong; Sangdon Jin

doi:10.3934/dcds.2022010

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2022, Volume 42, Issue 7: 3103-3118. Doi: 10.3934/dcds.2022010

This issue Previous Article Analytic linearization of a generalization of the semi-standard map: Radius of convergence and Brjuno sum Next Article Singular solutions of Toda system in high dimensions

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

Younghun Hong and
Sangdon Jin^,

Department of Mathematics, Chung-Ang University, Seoul 06974, Republic of Korea

^*Corresponding author: Sangdon Jin
^*Corresponding author: Sangdon Jin

Received: October 31, 2021

Published: July 2022

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Abstract

For the one-dimensional mass-critical and supercritical pseudo-relativistic nonlinear Schrödinger 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 [2]. 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.

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Mathematics Subject Classification: Primary: 35J10; Secondary: 35J61.

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