Bifurcation in a multi-component system of nonlinear Schrödinger equations

Abstract

We consider the system -Δu_j + a(x)u_j = μ_j u_j^3 + \be\sum_{k\ne j}u_k^2u_j, u_j>0, \qquad j=1,...,n, on a possibly unbounded domain $\Om\subset\R^N$, $N\le 3$, with Dirichlet boundary conditions. The system appears in nonlinear optics and in the analysis of mixtures of Bose-Einstein condensates. We consider the self-focussing (attractive self-interaction) case ${\mu }_1,...,{\mu }_n>0$ and take $\be\in\R$ as bifurcation parameter. There exists a branch of positive solutions with $u_j/u_k$ being constant for all $j,k\in 1,...,n$. The main results are concerned with the bifurcation of solutions from this branch. Using a hidden symmetry we are able to prove global bifurcation even when the linearization has even-dimensional kernel (which is always the case when $n>1$ is odd).

17 pages

References in corpus (1)

• Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition

Cited by in corpus (2)

• Bifurcations for a Coupled Schrödinger System with Multiple Components
• On a two-component Bose-Einstein condensate with steep potential wells