Stable solitary waves with prescribed $L^2$-mass for the cubic Schr\"odinger system with trapping potentials

TL;DR

This paper studies the existence and stability of standing wave solutions with fixed mass in the cubic Schrödinger system, providing variational characterizations and conditions for orbital stability in various settings.

Contribution

It introduces a variational approach to characterize and analyze the stability of mass-prescribed standing waves in the cubic Schrödinger system with trapping potentials.

Findings

Existence of mass-prescribed standing waves in bounded domains and $\mathbb{R}^N$ for $N extless=3$.

Orbital stability conditions derived from Grillakis-Shatah-Strauss theory.

Stable solitary waves exist for small masses and in the defocusing case for any masses.

Abstract

For the cubic Schr\"odinger system with trapping potentials in $\mathbb{R}^N$, $N\leq3$, or in bounded domains, we investigate the existence and the orbital stability of standing waves having components with prescribed $L^2$-mass. We provide a variational characterization of such solutions, which gives information on the stability through of a condition of Grillakis-Shatah-Strauss type. As an application, we show existence of conditionally orbitally stable solitary waves when: a) the masses are small, for almost every scattering lengths, and b) in the defocusing, weakly interacting case, for any masses.