We study standing waves for a nonlinear Schrödinger equation on a star graph G, i.e. N halflines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength α≤0. The nonlinearity is of focusing power type. The dynamics is given by an equation of the form iddtΨt=HΨt-|Ψt|2μΨt, where H is the Hamiltonian operator which generates the linear Schrödinger dynamics. We show the existence of several families of standing waves for every sign of the coupling at the vertex for every ω>α2N2. 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 if the nonlinearity is subcritical or critical, and for ω<ω* otherwise
We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two infinite, straight edges, i.e. a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a suitable sense and we obtain two possible limits: the Laplacian on the line with Dirichlet boundary conditions in the origin and a non trivial family of point perturbations of the Laplacian on the line. The first case generically occurs and corresponds to the decoupling of the two components of the limit curve, while in the second case a coupling takes place. We present also two families of curves which give rise to coupling.Comment: 20 pages, no figures. Minor misprint corrected, two references adde
We define the Schrödinger 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 and δ 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 [17] 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.
Abstract. We consider a nonlinear Schrödinger equation with focusing nonlinearity of power type on a star graph G, written as i∂ t Ψ(t) = HΨ(t) − |Ψ(t)| 2µ Ψ(t) , where H is the selfadjoint operator which defines the linear dynamics on the graph with an attractive δ interaction, with strength α < 0, at the vertex. The mass and energy functionals are conserved by the flow. We show that for 0 < µ < 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 certainΨ m ∈ H 1 (G), while for m > m * there is no minimum. Moreover, the set of minimizers has the structure M = {e iθΨ m , θ ∈ R}. Correspondingly, for every m < m * there exists a unique ω = ω(m) such that the standing waveΨ ω e iωt 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 α = 0.
In distinction to the Neumann case the squeezing limit of a Dirichlet network leads in the threshold region generically to a quantum graph with disconnected edges, exceptions may come from threshold resonances. Our main point in this paper is to show that modifying locally the geometry we can achieve in the limit a nontrivial coupling between the edges including, in particular, the class of δ-type boundary conditions. We work out an illustration of this claim in the simplest case when a bent waveguide is squeezed.A proper understanding of the problem requires, however, to find an interpretation of the coupling in terms of models without free parameters; a natural idea is to investigate the motion of a free quantum particle on a system of thin tubes which shrink towards the graph. This is a longstanding and nontrivial problem and the answer depends substantially on the network dynamics we work with. In the case when the Hamiltonian (which can be identified with the Laplacian by a suitable choice of units) refers to tubes with Neumann boundary, the limit yields typically quantum graphs with free boundary conditions (often called not quite appropriately Kirchhoff) in the vertices, described by U = 2 n J − I, where J is the n × n matrix whose all entries are equal to one -see [FW93, KZ01, RS01, Sa01], the same is true also for shrinking families of "sleeve" manifolds without a boundary [EP05]; the convergence is norm-resolvent [Po05a] and the conclusion extends to resonances on such structures [EP07].From the viewpoint of application to semiconductor structures and similar objects, however, it is the case of Dirichlet (hard-wall) boundary which is more important. It is very different from its Neumann counterpart and more difficult and relevant results started appearing only recently. The main source of the difficulties is that in the Dirichlet case geometric perturbations like bending, "swelling", twisting or branching give rise to an effective interaction -see, e.g., [EŠ89, DE95, EEK05] and references therein -sometimes attractive, sometimes repulsive, which changes spectral and scattering properties of such networks. Even the statement of the problem is more complicated than in the Neumann case where we naturally investigate spectrum around the zero value which is the continuum threshold. The analogous quantity in a Dirichlet network blows up to infinity and one has to choose the reference point by a suitable energy renormalization. In most cases, with the notable exception of the recent paper [MV06], the attention is paid to the vicinity of the "running" threshold.It is generally conjectured, that the generic limit in the vicinity of the threshold corresponds to a fully decoupled graph with Dirichlet conditions at the edge endpoints. This is obvious if the vertex regions squeeze faster than tubes as in [Po05b] but it is expected to hold even without such an additional effective repulsion. To see the reason one has to realize that the problem at hand can be by scaling rephrased as analysis of a network of tubes of ...
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