We consider the nonlinear Schrödinger equation in R d i∂tu + ∆u + f (u) = 0.For d 2, this equation admits travelling wave solutions of the form e iωt Φ(x) (up to a Galilean transformation), where Φ is a fixed profile, solution to −∆Φ + ωΦ = f (Φ), 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. N j=1 R j (t, x).

We study the stability of the cnoidal, dnoidal and snoidal elliptic functions as spatially-periodic standing wave solutions of the 1D cubic nonlinear Schrödinger equations. First, we give global variational characterizations of each of these periodic waves, which in particular provide alternate proofs of their orbital stability with respect to same-period perturbations, restricted to certain subspaces. Second, we prove the spectral stability of the cnoidal waves (in a certain parameter range) and snoidal waves against same-period perturbations, thus providing an alternate proof of this (known) fact, which does not rely on complete integrability. Third, we give a rigorous version of a formal asymptotic calculation of Rowlands to establish the instability of a class of real-valued periodic waves in 1D, which includes the cnoidal waves of the 1D cubic focusing nonlinear Schrödinger equation, against perturbations with period a large multiple of their fundamental period. Finally, we develop a numerical method to compute the minimizers of the energy with fixed mass and momentum constraints. Numerical experiments support and complete our analytical results.