Normalized solutions for Nonlinear Schr\"odinger systems on bounded domains

TL;DR

This paper investigates the existence and stability of normalized solutions to nonlinear Schrödinger systems on bounded domains, covering subcritical, critical, supercritical, and Sobolev-critical cases, with implications for physical models.

Contribution

It provides new sufficient conditions for the existence and stability of normalized solutions in various critical regimes, including the Sobolev-critical case.

Findings

Existence of orbitally stable standing waves in subcritical and critical cases.

Identification of local minimizers in supercritical cases.

Extension of analysis to Sobolev-critical nonlinearities.

Abstract

We analyze $L^2$-normalized solutions of nonlinear Schr\"odinger systems of Gross-Pitaevskii type, on bounded domains, with homogeneous Dirichlet boundary conditions. We provide sufficient conditions for the existence of orbitally stable standing waves. Such waves correspond to global minimizers of the associated energy in the $L^2$-subcritical and critical cases, and to local ones in the $L^2$-supercritical case. Notably, our study includes also the Sobolev-critical case.