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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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