On Asymptotic Stability in Energy Space of Ground States for Nonlinear Schrödinger Equations

Abstract

We consider nonlinear Schrödinger equationswhere d ≥ 3 and β is smooth. We prove that symmetric finite energy solutions close to orbitally stable ground states converge to a sum of a ground state and a dispersive wave as t → ∞ assuming the so called Fermi Golden Rule (FGR) hypothesis. We improve the "sign condition"required in a recent paper by Gang Zhou and I.M.Sigal.

On instability for the quintic nonlinear Schrodinger equation of some approximate periodic solutions

“…Once the necessary spectral hypotheses in [CM,Cu4] are proved in Section 3, Theorem 1.1 is a direct consequence of [CM,Cu4]. Nonetheless, we give a sketch of the main steps in the proof.…”

“…SET-UP FOR THEOREM 1.1 AND DISPERSION FOR THE LINEARIZATION Theorem 1.1 is a consequence of [Cu3,Cu4]. Notice that since the linearization has just one pair of nonzero eigenvalues, the Hamiltonian set up in [Cu1] is unnecessary, and the theory in [Cu3,Cu4,CM] is adequate. We recall that due to the absence of the endpoint Strichartz estimate in 1D, the theory requires some adequate surrogate.…”

“…Here we repeat the theory in [CM,Cu4], which is somewhat more elementary than [Cu1], but still adequate in our setting.…”

“…In the present paper, though, we treat a situation that is quite special because of the hypothesis that σ d (H L ) consists of just two eigenvalues. Hence, it is enough to use the simpler framework of [CM,Cu4] (we recall that [Cu4] is a revision and a simplification of [Cu3], which contains some errors).…”

“…It is plausible that, generically, they should scatter via an asymptotic stability analysis to an asymmetric nonlinear bound state or to the 0 solution. Proving this can be a challenging problem, because even if the solutions in [MW] scatter to a ground state, they may go through intermediate phases which are not within the reach of the local analysis near a ground state such as that in [CM,Cu1]. The issues then seem in some sense more "global", and closer in spirit to the problems addressed in [TY,SW3].…”