Paper detail

Orbital and asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three

We begin to study in this paper orbital and asymptotic stability of standing waves for a model of Schrödinger 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 selfadjoint operator $H_α$, where the strength $α$ depends on the wavefunction: $i\dot u= H_αu$, $α=α(u)$. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of its singular part, 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 with concentrated nonlinearity of power type. For such a model we prove the existence of standing waves of the form $u (t)=e^{iωt}Φ_ω$, which are orbitally stable in the range $σ\in (0,1)$, and orbitally unstable for $σ\geq 1.$ Moreover, we show that for $σ\in (0,\frac{1}{\sqrt 2})$ 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 $L^p$ space which allows dispersive estimates, the following resolution holds: $u(t) = e^{iω_{\infty} t} Φ_{ω_{\infty}} +U_t*ψ_{\infty} +r_{\infty}, \quad \textrm{as} \;\; t \rightarrow +\infty$, where $U$ is the free Schrödinger propagator, $ω_{\infty} > 0$ and $ψ_{\infty}$, $r_{\infty} \in L^2(\R^3)$ with $\| r_{\infty} \|_{L^2} = O(t^{-5/4}) \quad \textrm{as} \;\; t \rightarrow +\infty$. Notice that in the present model the admitted nonlinearity for which asymptotic stability of solitons is proved is subcritical.