Existence and stability of standing waves for coupled nonlinear Hartree type equations

Abstract

We study existence and stability of standing waves for coupled nonlinear Hartree type equations \[ -i\frac{\partial}{\partial t}ψ_j=Δψ_j+\sum_{k=1}^m \left(W\star |ψ_k|^p \right)|ψ_j|^{p-2}ψ_j, \] where ${\psi }_j:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{C}$ for $j=1,\dots ,m$ and the potential $W:\mathbb{R}\to [0,\infty )$ satisfies certain assumptions. Our method relies on a variational characterization of standing waves based on minimization of the energy when $L^2$ norms of component waves are prescribed. We obtain existence and stability results for two and three-component systems and for a certain range of $p$. In particular, our argument works in the case when $W(x)=|x{|}^{-\alpha }$ for some $\alpha >0.$