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=0x=0 obtained considering a contact (or δδ) interaction with strength αα, which consists of a singular perturbation of the Laplacian described by a selfadjoint operator HαHα, and letting the strength αα depend on the wavefunction in a prescribed way: i˙u=Hαuiu˙=Hαu, α=α(u)α=α(u). For power nonlinearities in the range (1√2,1)(12,1) there exist orbitally stable standing waves ΦωΦω, and the linearization around them admits two imaginary eigenvalues (neutral modes, absent in the range (0,1√2)(0,12) 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∗)(12,σ∗) for a certain σ∗∈(1√2,√3+12√2]σ∗∈(12,3+122], the dynamics near the orbit of a standing wave asymptotically relaxes in the following sense: consider an initial datum u(0)u(0), suitably near the standing wave Φω0,Φω0, then the solution u(t)u(t) can be asymptotically decomposed asu(t)=eiω∞t+ib1log(1+ϵk∞t)+iγ∞Φω∞+Ut∗ψ∞+r∞,ast→+∞,u(t)=eiω∞t+ib1log⁡(1+ϵk∞t)+iγ∞Φω∞+Ut∗ψ∞+r∞,ast→+∞,where ω∞ω∞, k∞,γ∞>0k∞,γ∞>0, b1∈Rb1∈R, and ψ∞ψ∞ and r∞∈L2(R3)r∞∈L2(R3) , U(t)U(t) is the free Schrödinger group and∥r∞∥L2=O(t−1/4)ast→+∞ .‖r∞‖L2=O(t−1/4)ast→+∞ .We stress the fact that in the present case and contrarily to the main results in the field, the admitted nonlinearity is L2L2-subcritical.