Asymptotic stability for standing waves of a NLS equation with subcritical concentrated nonlinearity in dimension three: Neutral modes

In this paper the study of asymptotic stability of standing waves for a model of Schrödinger equation with spatially concentrated nonlinearity in dimension three. The nonlinearity studied is a power nonlinearity concentrated at the point x = 0 obtained considering a contact (or δ) interaction with strength α, which consists of a singular perturbation of the Laplacian described by a selfadjoint operator Hα, and letting the strength α depend on the wave-function in a prescribed way: iu̇ = Hαu, α = α(u). For power nonlinearities in the range (1/√2, 1) there exist orbitally stable standing waves Φω, and the linearization around them admits two imaginary eigenvalues (neutral modes, absent in the range (0, 1/√2) previously treated by the same authors) which in principle could correspond to non decaying states, so preventing asymptotic relaxation towards an equilibrium orbit. We prove that, in the range (1/√2, σ∗) for a certain σ∗ ∈ (1/√2, √3+1/2√2], the dynamics near the orbit of a standing wave asymptotically relaxes in the following sense: consider an initial datum u(0), suitably near the standing wave Φω0, then the solution u(t) can be asymptotically decomposed as u(t) = eiω∞t+ib1 log(1+∈k∞t)+iγ∞Φω∞ + Ut ∗ ψ∞ + r∞, as t → +∞, where ω∞, k∞, γ∞ > 0, b1 ∈ ℝ, and ψ∞ and r∞ ∈ L2(ℝ3) , U(t) is the free Schrödinger group and ∥r∞∥L2 = O(t-1/4) as t → +∞. We stress the fact that in the present case and contrarily to the main results in the field, the admitted nonlinearity is L2-subcritical.