## Asymptotic stability for standing waves of a NLS equation with subcritical concentrated nonlinearity in dimension three: neutral modes.(English)Zbl 1351.35187

Summary: 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 $u(t) = e^{i\omega_{\infty}t+ib_{1}\log(1 +\epsilon k_{\infty}t)+i\gamma_{\infty}} \Phi_{\omega_{\infty}}+U_t\ast\psi_{\infty}+r_{\infty},\quad\,\,\text{as}\,\, t \rightarrow +\infty,$ 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 $\|r_{\infty}\|_{L^{2}} = O(t^{-1/4})\quad\,\,\text{as}\,\, t \rightarrow +\infty\;.$ 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.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35Q51 Soliton equations 37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems 35B40 Asymptotic behavior of solutions to PDEs 35B35 Stability in context of PDEs
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