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

Abstract

In this paper the study of asymptotic stability ofstanding waves for a model of Schrödinger equation with spatially concentratednonlinearity 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 Laplaciandescribed by a selfadjoint operator $H_{\alpha}$, and letting the strength $\alpha$ depend on the wavefunction in a prescribed way: $i\dot u= H_\alphau$, $\alpha=\alpha(u)$. 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 imaginaryeigenvalues (neutral modes, absent in the range $(0,\frac{1}{\sqrt 2})$ previouslytreated by the same authors) which in principle could correspond to non decaying states,so preventing asymptotic relaxation towards an equilibrium orbit. We prove that, inthe 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 mainresults in the field, the admitted nonlinearity is $L^2$-subcritical.