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

Riccardo Adami; Diego Noja; Cecilia  Ortoleva

doi:10.3934/dcds.2016057

Article Contents

2016, Volume 36, Issue 11: 5837-5879. Doi: 10.3934/dcds.2016057

This issue Previous Article Next Article Existence of positive multi-bump solutions for a Schrödinger-Poisson system in $\mathbb{R}^{3}$

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

1.
Dipartimento di Scienze Matematiche, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino
2.
Dipartimento di Matematica e Applicazioni, Università di Milano Bicocca, via R. Cozzi 53, 20125 Milano
3.
PriceWaterhouseCoopers Italia, Via Monte Rosa 91, 21049, Milano

Received: September 2015

Revised: May 2016

Published online: August 2016

Abstract / Introduction Related Papers Cited by

Abstract

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 $\delta$) interaction with strength $\alpha$, which consists of a singular perturbation of the Laplacian described by a selfadjoint operator $H_{\alpha}$, and letting the strength $\alpha$ depend on the wavefunction in a prescribed way: $i\dot u= H_\alpha u$, $\alpha=\alpha(u)$. For power nonlinearities in the range $(\frac{1}{\sqrt 2},1)$ there exist orbitally stable standing waves $\Phi_\omega$, and the linearization around them admits two imaginary eigenvalues (neutral modes, absent in the range $(0,\frac{1}{\sqrt 2})$ 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 $(\frac{1}{\sqrt 2},\sigma^*)$ for a certain $\sigma^* \in (\frac{1}{\sqrt{2}}, \frac{\sqrt{3} +1}{2 \sqrt{2}}]$, 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 $\Phi_{\omega_0}, $ then the solution $u(t)$ can be asymptotically decomposed as $$u(t) = e^{i\omega_{\infty} t +i b_1 \log (1 +\epsilon k_{\infty} t) + i \gamma_\infty} \Phi_{\omega_{\infty}} +U_t*\psi_{\infty} +r_{\infty}, \quad \textrm{as} \;\; t \rightarrow +\infty,$$ where $\omega_{\infty}$, $k_{\infty}, \gamma_\infty > 0$, $b_1 \in \mathbb{R}$, and $\psi_{\infty}$ and $r_{\infty} \in L^2(\mathbb{R}^3)$ , $U(t)$ is the free Schrödinger group and $$\| r_{\infty} \|_{L^2} = O(t^{-1/4}) \quad \textrm{as} \;\; t \rightarrow +\infty\ .$$ We stress the fact that in the present case and contrarily to the main results in the field, the admitted nonlinearity is $L^2$-subcritical.

Keywords:

Nonlinear equations of Schrödinger type,
point interactions,
standing waves,
asymptotic stability.

Mathematics Subject Classification: Primary: 35Q55, 37Q51, 37K40.

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