Infinitely many new solutions for a nonlinear coupled Schr\"odinger system

TL;DR

This paper proves the existence of infinitely many new solutions to a coupled nonlinear Schrödinger system, including synchronized and segregated types, with detailed analysis of their structure and non-degeneracy.

Contribution

It introduces a novel class of solutions for coupled Schrödinger systems, employing finite dimensional reduction and proving their non-degeneracy.

Findings

Existence of infinitely many synchronized solutions.

Existence of segregated solutions concentrating near infinity.

Solutions are non-degenerate, enabling further analysis.

Abstract

We revisit the following nonlinear Schr\"odinger system \begin{align*}\begin{cases} -\epsilon^{2}\Delta u +P(x) u= \mu_1 u^3 +\beta uv^2, &~\text{in}\;\mathbb {R}^3,\\ -\epsilon^{2}\Delta v+Q(x) v= \mu_2 v^3 +\beta u^2v, &~\text{in}\;\mathbb{ R}^3, \end{cases} \end{align*} where $\epsilon$ is a positive parameter, $P(x),\,Q(x)$ are the potential functions, $\mu_1>0$, $\mu_2>0$ and $\beta\in\mathbb R$ is a coupling constant. Employing the finite dimensional reduction method, we prove that there are new kind of synchronized and segregated solutions, which concentrate both in a bounded domain and near infinity, and present a special structure. Moreover, by applying the local Pohozaev identities and some point-wise estimates of the errors, we prove that the new kind of synchronized solutions are non-degenerate, which is of great interest independently. One of the main difficulties of Schr\"odinger system come from the interspecies interaction between the components, which never appear in the study of single equation. Secondly, prior to the construction of new solutions, we shall verify the non-degeneracy of the solutions established in [Peng-Pi, Discrete Contin. Dyn. Syst., 2016] for the Schr\"odinger systems.