Studies on a system of nonlinear Schrödinger equations with potential and quadratic interaction
Vicente Alvarez, Amin Esfahani
TL;DR
This work analyzes the existence and multiplicity of normalized standing waves for a coupled nonlinear Schrödinger system with quadratic interaction and trapping potentials. By formulating a variational problem with energy functional $I(\phi,\psi)$ under $L^2$-type constraints, the authors deploy profile decomposition and concentration-compactness to overcome noncompactness introduced by the potentials. They establish ground-state local minimizers for the harmonic and partial confinement settings, prove the existence of a second normalized (Mountain-Pass) solution, and characterize the asymptotic behavior of minimizers in terms of the lowest eigenfunctions of the related linear operators, including a rigorous link to a one-dimensional reduced system. The results illuminate how spectral properties of the linear part govern the existence, structure, and asymptotics of standing waves in high and low dimensions, and they provide a framework for further study of long-time dynamics and blow-up phenomena in this coupled NLS setting.
Abstract
In this work, we study the existence of various classes of standing waves for a nonlinear Schrödinger system with quadratic interaction, along with a harmonic or partially harmonic potential. We establish the existence of ground-state normalized solutions for this system, which serve as local minimizers of the associated functionals. To address the difficulties raised by the potential term, we employ profile decomposition and concentration-compactness principles. The absence of global energy minimizers in critical and supercritical cases leads us to focus on local energy minimizers. Positive results arise in scenarios of partial confinement, attributed to the spectral properties of the associated linear operators. Furthermore, we demonstrate the existence of a second normalized solution using Mountain-pass geometry, effectively navigating the difficulties posed by the nonlinear terms. We also explore the asymptotic behavior of local minimizers, revealing connections with unique eigenvectors of the linear operators. Additionally, we identify global and blow-up solutions over time under specific conditions, contributing new insights into the dynamics of the system.