High-speed excited multi-solitons in nonlinear Schrodinger equations

We consider the nonlinear Schrodinger equation in R-d i partial derivative(t)u + Delta u + f(u) = 0. For d >= 2. this equation admits traveling wave solutions of the form e(i omega t)Phi(x) (up to a Galilean transformation), where Phi is a fixed profile, solution to -Delta Phi + omega Phi = f(Phi), but not the ground state. This kind of profiles are called excited states. In this paper, we construct solutions to NLS behaving like a sum of N excited states which spread up quickly as time grows (which we call multi-solitons). We also show that if the flow around one of these excited states is linearly unstable, then the multi-soliton is not unique, and is unstable. (C) 2011 Elsevier Masson SAS. All rights reserved.