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

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.