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

Abstract

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 $\alpha$, which consists of a singular perturbation of the laplacian described by a selfadjoint operator $H_{\alpha }$, where the strength $\alpha$ depends on the wavefunction: $i\dot{u}=H_{\alpha }u$, $\alpha =\alpha (u)$. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of its singular part, we let the strength $\alpha$ depend on $u$ according to the law $\alpha =-\nu |q{|}^{\sigma }$, with $\nu >0$. 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 $u(t)=e^{i\omega t}{\Phi }_{\omega }$, which are orbitally stable in the range $\sigma \in (0,1)$, and orbitally unstable for $\sigma \ge 1.$ Moreover, we show that for $\sigma \in (0,\frac{1}{\sqrt{2}})$ 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: $u(t)=e^{i{\omega }_{\infty }t}{\Phi }_{{\omega }_{\infty }}+U_t\ast {\psi }_{\infty }+r_{\infty },\text{as}t\to +\infty$, where $U$ is the free Schrödinger propagator, ${\omega }_{\infty }>0$ and ${\psi }_{\infty }$, $r_{\infty }\in L^2({\mathbb{R}}^3)$ with $\Vert r_{\infty }{\Vert }_{L^2}=O(t^{-5/4})\text{as}t\to +\infty$. Notice that in the present model the admitted nonlinearity for which asymptotic stability of solitons is proved is subcritical.

Comments and clarifications added; several misprints corrected

References in corpus (2)

• Solitary Wave Dynamics in an External Potential
• Solitons supported by localized nonlinearities in periodic media

Cited by in corpus (16)

• Soliton transport in tubular networks: transmission at vertices in the shrinking limit
• Ground states for the planar NLSE with a point defect as minimizers of the constrained energy
• Sine-Gordon solitons in networks: Scattering and transmission at vertices
• Dynamics of Dirac solitons in networks
• Existence, structure, and robustness of ground states of a NLSE in 3D with a point defect
• The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit
• The point-like limit for a NLS equation with concentrated nonlinearity in dimension three
• Attractors of Hamilton nonlinear partial differential equations
• Global attractor for 1D Dirac field coupled to nonlinear oscillator
• On Global attraction to solitary waves for Klein-Gordon equation with concentrated nonlinearity
• Stability of the standing waves of the concentrated NLSE in dimension two
• A general review on the NLS equation with point-concentrated nonlinearity
• On global well-posedness for Klein-Gordon equation with concentrated nonlinearity
• Well-posedness of the three-dimensional NLS equation with sphere-concentrated nonlinearity
• Microscopic derivation of a Schrödinger equation in dimension one with a nonlinear point interaction
• Normalized solutions of one-dimensional defocusing NLS equations with nonlinear point interactions