The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit

In the present paper we study the following scaled nonlinear Schr\"odinger equation (NLS) in one space dimension: $$i\frac{d}{dt} \psi^{\varepsilon}(t) =-\Delta\psi^{\varepsilon}(t) + \frac{1}{\epsilon}V\left(\frac{x}{\epsilon}\right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t) \quad \quad \epsilon>0\ ,\quad V\in L^1(\mathbb{R},(1+|x|)dx) \cap L^\infty(\mathbb{R}) \ .$$ This equation represents a nonlinear Schr\"odinger equation with a spatially concentrated nonlinearity. We show that in the limit $\epsilon\to 0$, the weak (integral) dynamics converges in $H^1(\mathbb{R})$ to the weak dynamics of the NLS with point-concentrated nonlinearity: $$i\frac{d}{dt} \psi(t) =H_{\alpha}\psi(t) .$$ where $H_{\alpha}$ is the laplacian with the nonlinear boundary condition at the origin $\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)$ and $\alpha=\int_{\mathbb{R}}Vdx$. The convergence occurs for every $\mu\in \mathbb{R}^+$ if $V \geq 0$ and for every $\mu\in (0,1)$ otherwise. The same result holds true for a nonlinearity with an arbitrary number $N$ of concentration pointsComment: 10 page

Constrained energy minimization and orbital stability for the NLS equation on a star graph

We consider a nonlinear Schr\"odinger equation with focusing nonlinearity of power type on a star graph ${\mathcal G}$, written as $i \partial_t \Psi (t) = H \Psi (t) - |\Psi (t)|^{2\mu}\Psi (t)$, where $H$ is the selfadjoint operator which defines the linear dynamics on the graph with an attractive $\delta$ interaction, with strength $\alpha < 0$, at the vertex. The mass and energy functionals are conserved by the flow. We show that for $0<\mu<2$ the energy at fixed mass is bounded from below and that for every mass $m$ below a critical mass $m^*$ it attains its minimum value at a certain \hat \Psi_m \in H^1(\GG) , while for $m>m^*$ there is no minimum. Moreover, the set of minimizers has the structure {\mathcal M}={e^{i\theta}\hat \Psi_m, \theta\in \erre}. Correspondingly, for every $m<m^*$ there exists a unique $\omega=\omega(m)$ such that the standing wave $\hat\Psi_{\omega}e^{i\omega t}$ is orbitally stable. To prove the above results we adapt the concentration-compactness method to the case of a star graph. This is non trivial due to the lack of translational symmetry of the set supporting the dynamics, i.e. the graph. This affects in an essential way the proof and the statement of concentration-compactness lemma and its application to minimization of constrained energy. The existence of a mass threshold comes from the instability of the system in the free (or Kirchhoff's) case, that in our setting corresponds to \al=0.Comment: 26 pages, 1 figur

Variational properties and orbital stability of standing waves for NLS equation on a star graph

We study standing waves for a nonlinear Schr\"odinger equation on a star graph {$\mathcal{G}$} i.e. $N$ half-lines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength $\alpha\leqslant 0$. The nonlinearity is of focusing power type. The dynamics is given by an equation of the form $i \frac{d}{dt}\Psi_t = H \Psi_t - | \Psi_t |^{2\mu} \Psi_t$, where $H$ is the Hamiltonian operator which generates the linear Schr\"odinger dynamics. We show the existence of several families of standing waves for every sign of the coupling at the vertex for every $\omega > \frac{\alpha^2}{N^2}$. Furthermore, we determine the ground states, as minimizers of the action on the Nehari manifold, and order the various families. Finally, we show that the ground states are orbitally stable for every allowed $\omega$ if the nonlinearity is subcritical or critical, and for $\omega<\omega^\ast$ otherwise.Comment: 36 pages, 2 figures, final version appeared in JD

On the Asymptotic Dynamics of a Quantum System Composed by Heavy and Light Particles

We consider a non relativistic quantum system consisting of $K$ heavy and $N$ light particles in dimension three, where each heavy particle interacts with the light ones via a two-body potential $\alpha V$. No interaction is assumed among particles of the same kind. Choosing an initial state in a product form and assuming $\alpha$ sufficiently small we characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. In the case K=1 the result is extended to arbitrary $\alpha$. The proof relies on a perturbative analysis and exploits a generalized version of the standard dispersive estimates for the Schr\"{o}dinger group. Exploiting the asymptotic formula, it is also outlined an application to the problem of the decoherence effect produced on a heavy particle by the interaction with the light ones.Comment: 38 page

Fast solitons on star graphs

We define the Schr\"odinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global well-posedness in the energy domain and conservation laws for some self-adjoint boundary conditions at the vertex, i.e. Kirchhoff boundary condition and the so called $\delta$ and $\delta'$ boundary conditions. Moreover, in the same setting we study the collision of a fast solitary wave with the vertex and we show that it splits in reflected and transmitted components. The outgoing waves preserve a soliton character over a time which depends on the logarithm of the velocity of the ingoing solitary wave. Over the same timescale the reflection and transmission coefficients of the outgoing waves coincide with the corresponding coefficients of the linear problem. In the analysis of the problem we follow ideas borrowed from the seminal paper \cite{[HMZ07]} about scattering of fast solitons by a delta interaction on the line, by Holmer, Marzuola and Zworski; the present paper represents an extension of their work to the case of graphs and, as a byproduct, it shows how to extend the analysis of soliton scattering by other point interactions on the line, interpreted as a degenerate graph.Comment: Sec. 2 revised; several misprints corrected; added references; 32 page

Blow-up and instability of standing waves for the NLS with a point interaction in dimension two

In the present note we study the NLS equation in dimension two with a point interaction and in the supercritical regime, showing two results. After obtaining the (nonstandard) virial formula, we exhibit a set of initial data that blow-up. Moreover we show the standing waves $e^{i\omega t} \varphi_\omega$ corresponding to ground states $\varphi_\omega$ of the action are strongly unstable, at least for sufficiently high $\omega$.Comment: 12 pages, minor modification