Orbital and asymptotic stability for standing waves of a nonlinear Schro?dinger equation with concentrated nonlinearity in dimension three

We study orbital and asymptotic stability of standing waves for a model of Schr¨odinger equation with concentrated nonlinearity in dimension three. The nonlinearity is obtained considering a point (or contact) interaction with strength α, which consists of a singular perturbation of the Laplacian described by a self-adjoint operator Hα, and letting the strength α depend on the wavefunction: i ˙ u = Hαu, α =α(u). It is well-known that the elements of the domain of such operator can be written as the sum of a regular function and a function that exhibits a singularity proportional to |x − x0|−1, where x0 is the location of the point interaction. If q is the so-called charge of the domain element u, i.e., the coefficient of its singular part, then, in order to introduce a nonlinearity, we let the strength α depend on u according to the law α = −ν|q|σ , with ν > 0. This characterizes the model as a focusing NLS (nonlinear Schr¨odinger) with concentrated nonlinearity of power type. For such a model we prove the existence of standing waves of the form u(t) =eiωtω,which are orbitally stable in the range σ ∈(0, 1), and orbitally unstablewhen σ ≥ 1.Moreover, we show that for σ ∈ (0, √1 2 ) every standing wave is asymptotically stable in the following sense. Choosing initial data close to the stationary state in the energy norm, and belonging to a natural weighted Lp space which allows dispersive estimates, the following resolution holds: u(t) = eiω∞tω∞ + Ut ∗ ψ∞ + r∞, where U is the free Schr¨odinger propagator, ω∞ > 0 and ψ∞, r∞ ∈ L2(R3) with r∞L2 = O(t−5/4) as t →+∞.Notice that in the present model the admitted nonlinearity for which asymptotic stability of solitons is proved is subcritical, in the sense that it does not give rise to blow up, regardless of the chosen initial data.