Segregated Vector Solutions for linearly coupled Nonlinear Schr\"odinger Systems

TL;DR

This paper constructs non-radial segregated vector solutions with multiple bumps for linearly coupled nonlinear Schrödinger systems in three dimensions, analyzing the impact of linear coupling on solution structure.

Contribution

It introduces a method to explicitly construct multi-bump, non-radial solutions for coupled Schrödinger systems, revealing how linear coupling influences solution segregation.

Findings

Existence of non-radial vector solutions with exactly  positive bumps.

Solutions are constructed for small positive coupling constants.

Explicit characterization of the solution features is provided.

Abstract

We consider the following system linearly coupled by nonlinear Schr\"odinger equations in $\R^3$ $$ \left\{\begin{array}{ll} -\Delta u_j+u_j=u^3_j-\va\sum\limits_{i\neq j}^N u_i,\{1cm}& x\in \R^3, \{0.2cm}\\ u_j\in H^1(\R^3),\quad j=1,\cdots,N, \end{array} \right. $$ where $\va\in\R$ is a coupling constant. This type of system arises in particular in models in nonlinear $N$-core fiber. We examine the effect of the linear coupling to the solution structure. When $N=2,3$, for any prescribed integer $\ell\ge 2$, we construct a non-radial vector solutions of segregated type, with two components having exactly $\ell$ positive bumps for $\va>0$ sufficiently small. We also give an explicit description on the characteristic features of the vector solutions.