67 research outputs found

    Wave equation with concentrated nonlinearities

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    In this paper we address the problem of wave dynamics in presence of concentrated nonlinearities. Given a vector field VV on an open subset of \CO^n and a discrete set Y\subset\RE^3 with nn elements, we define a nonlinear operator ΔV,Y\Delta_{V,Y} on L^2(\RE^3) which coincides with the free Laplacian when restricted to regular functions vanishing at YY, and which reduces to the usual Laplacian with point interactions placed at YY when VV is linear and is represented by an Hermitean matrix. We then consider the nonlinear wave equation ϕ¨=ΔV,Yϕ\ddot \phi=\Delta_{V,Y}\phi and study the corresponding Cauchy problem, giving an existence and uniqueness result in the case VV is Lipschitz. The solution of such a problem is explicitly expressed in terms of the solutions of two Cauchy problem: one relative to a free wave equation and the other relative to an inhomogeneous ordinary differential equation with delay and principal part ζ˙+V(ζ)\dot\zeta+V(\zeta). Main properties of the solution are given and, when YY is a singleton, the mechanism and details of blow-up are studied.Comment: Revised version. To appear in Journal of Physics A: Mathematical and General, special issue on Singular Interactions in Quantum Mechanics: Solvable Model

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

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    We study standing waves for a nonlinear Schr\"odinger equation on a star graph {G\mathcal{G}} i.e. NN half-lines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength α0\alpha\leqslant 0. The nonlinearity is of focusing power type. The dynamics is given by an equation of the form iddtΨt=HΨtΨt2μΨt i \frac{d}{dt}\Psi_t = H \Psi_t - | \Psi_t |^{2\mu} \Psi_t , where HH 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 ω>α2N2\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 ω\omega if the nonlinearity is subcritical or critical, and for ω<ω\omega<\omega^\ast otherwise.Comment: 36 pages, 2 figures, final version appeared in JD

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

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    We consider a nonlinear Schr\"odinger equation with focusing nonlinearity of power type on a star graph G{\mathcal G}, written as itΨ(t)=HΨ(t)Ψ(t)2μΨ(t) i \partial_t \Psi (t) = H \Psi (t) - |\Psi (t)|^{2\mu}\Psi (t), where HH is the selfadjoint operator which defines the linear dynamics on the graph with an attractive δ\delta interaction, with strength α<0\alpha < 0, at the vertex. The mass and energy functionals are conserved by the flow. We show that for 0<μ<20<\mu<2 the energy at fixed mass is bounded from below and that for every mass mm below a critical mass mm^* it attains its minimum value at a certain \hat \Psi_m \in H^1(\GG) , while for m>mm>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<mm<m^* there exists a unique ω=ω(m)\omega=\omega(m) such that the standing wave Ψ^ωeiωt\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

    Rigorous Dynamics and Radiation Theory for a Pauli-Fierz Model in the Ultraviolet Limit

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    The present paper is devoted to the detailed study of quantization and evolution of the point limit of the Pauli-Fierz model for a charged oscillator interacting with the electromagnetic field in dipole approximation. In particular, a well defined dynamics is constructed for the classical model, which is subsequently quantized according to the Segal scheme. To this end, the classical model in the point limit is reformulated as a second order abstract wave equation, and a consistent quantum evolution is given. This allows a study of the behaviour of the survival and transition amplitudes for the process of decay of the excited states of the charged particle, and the emission of photons in the decay process. In particular, for the survival amplitude the exact time behaviour is found. This is completely determined by the resonances of the systems plus a tail term prevailing in the asymptotic, long time regime. Moreover, the survival amplitude exhibites in a fairly clear way the Lamb shift correction to the unperturbed frequencies of the oscillator.Comment: Shortened version. To appear in J. Math. Phy

    Point interactions in acoustics: one dimensional models

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    A one dimensional system made up of a compressible fluid and several mechanical oscillators, coupled to the acoustic field in the fluid, is analyzed for different settings of the oscillators array. The dynamical models are formulated in terms of singular perturbations of the decoupled dynamics of the acoustic field and the mechanical oscillators. Detailed spectral properties of the generators of the dynamics are given for each model we consider. In the case of a periodic array of mechanical oscillators it is shown that the energy spectrum presents a band structure.Comment: revised version, 30 pages, 2 figure

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

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    In the present paper we study the following scaled nonlinear Schr\"odinger equation (NLS) in one space dimension: iddtψε(t)=Δψε(t)+1ϵV(xϵ)ψε(t)2μψε(t)ϵ>0 ,VL1(R,(1+x)dx)L(R) . 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 ϵ0\epsilon\to 0, the weak (integral) dynamics converges in H1(R)H^1(\mathbb{R}) to the weak dynamics of the NLS with point-concentrated nonlinearity: iddtψ(t)=Hαψ(t). i\frac{d}{dt} \psi(t) =H_{\alpha}\psi(t) . where HαH_{\alpha} is the laplacian with the nonlinear boundary condition at the origin ψ(t,0+)ψ(t,0)=αψ(t,0)2μψ(t,0)\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0) and α=RVdx\alpha=\int_{\mathbb{R}}Vdx. The convergence occurs for every μR+\mu\in \mathbb{R}^+ if V0V \geq 0 and for every μ(0,1)\mu\in (0,1) otherwise. The same result holds true for a nonlinearity with an arbitrary number NN of concentration pointsComment: 10 page

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

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    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 eiωtφωe^{i\omega t} \varphi_\omega corresponding to ground states φω\varphi_\omega of the action are strongly unstable, at least for sufficiently high ω\omega.Comment: 12 pages, minor modification

    A Dirac field interacting with point nuclear dynamics

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    The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent one-body Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions for data in H\u3c3 with \u3c3 08[1,32[. In the course of the analysis a second new result of independent interest is discussed and proved, namely the construction of the propagator for the Dirac operator with several moving Coulomb singularities

    Fast solitons on star graphs

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    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 δ\delta and δ\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
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