On coupled nonlinear Schrödinger systems with mixed couplings

Abstract:

We consider the following nonlinear Schrödinger system with mixed couplings in $\mathbb {R}^3$: \begin{equation*} -\Delta u_i + \lambda _i u_i=\mu _i u_i^3+\sum \limits _{j=1,j\neq i}^N\beta _{ij}u_j^2u_i,\ \ \ i=1,\cdots ,N, \end{equation*} where $\lambda _i, \mu _i>0, \beta _{ij}=\beta _{ji} (i,j=1,\cdots ,N, i\neq j)$. The system appears in modeling of Bose-Einstein condensates theory. While most existing works in the literature are concerned with purely attractive or purely repulsive couplings (i.e., all $\beta _{ij}$ have the same signs), we examine the effect of mixed nonlinear couplings on the solution structure and obtain vector solutions with some of the components synchronized between them while being segregated with the rest of the components simultaneously.