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∂∂tψj=Δψj+∑k=1mW⋆|ψk|p|ψj|p−2ψj, where ψj:RN×R→C for j = 1, …, m and W:R→[0,∞) satisfies certain assumptions. Our method relies on a variational characterization of standing waves based on minimization of the energy when L2 norms of component waves are prescribed. We obtain existence and stability results of standing waves for two and three-component coupled systems and for a certain range of p. In particular, our argument works in the case when W(x) = |x|−α for some α > 0.