Diego Noja

FAST SOLITONS ON STAR GRAPHS

We define the Schrodinger 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.

Nonlinear Schrödinger equation on graphs: recent results and open problems

In this paper, an introduction to the new subject of nonlinear dispersive Hamiltonian equations on graphs is given. The focus is on recently established properties of solutions in the case of the nonlinear Schrödinger (NLS) equation. Special consideration is given to the existence and behaviour of solitary solutions. Two subjects are discussed in some detail concerning the NLS equation on a star graph: the standing waves of the NLS equation on a graph with a δ interaction at the vertex, and the scattering of fast solitons through a Y-junction in the cubic case. The emphasis is on a description of concepts and results and on physical context, without reporting detailed proofs; some perspectives and more ambitious open problems are discussed.

On the structure of critical energy levels for the cubic focusing NLS on star graphs

We provide information on a non-trivial structure of phase space of the cubic nonlinear Schrodinger (NLS) on a three-edge star graph. We prove that, in contrast to the case of the standard NLS on the line, the energy associated with the cubic focusing Schrodinger equation on the three-edge star graph with a free (Kirchhoff) vertex does not attain a minimum value on any sphere of constant L2-norm. We moreover show that the only stationary state with prescribed L2-norm is indeed a saddle point.

Existence of dynamics for a 1D NLS equation perturbed with a generalized point defect

In the present paper we study the well-posedness for the one-dimensional cubic NLS perturbed by a generic point interaction. Point interactions are described as the 4-parameter family of self-adjoint extensions of the symmetric 1D Laplacian defined on the regular functions vanishing at a point, and in the present context can be interpreted as localized defects interacting with the NLS field. A previously treated special case is given by an NLS equation with a δ defect which we generalize and extend, as far as well-posedness is concerned, to the whole family of point interactions. We prove existence and uniqueness of the local Cauchy problem in strong form (initial data and evolution in the operator domain of point interactions), weak form (initial data and evolution in the form domain of point interactions) and . Conservation laws of mass and energy are proved for finite energy weak solutions of the problem, which imply global existence of the dynamics. A technical difficulty arises due to the fact that a power nonlinearity does not preserve the form domain for a subclass of point interactions; to overcome it, a technique based on the extension of resolvents of the linear part of the generator to maps between a suitable Hilbert space and the energy space is devised and estimates are given which show the needed regularization properties of the nonlinear flow.

Orbital and asymptotic stability for standing waves of a nonlinear Schrödinger equation with concentrated nonlinearity in dimension three

We begin to study in this paper orbital and asymptotic stability of standing waves for a model of Schrodinger equation with concentrated nonlinearity in dimension three. The nonlinearity is obtained considering a point (or contact) interaction with strength α, which consists of a singular perturbation of the Laplacian described by a self-adjoint operator Hα, and letting the strength α depend on the wavefunction: iu=Hαu, α = α(u). It is well-known that the elements of the domain of such operator can be written as the sum of a regular function and a function that exhibits a singularity proportional to |x − x0|−1, where x0 is the location of the point interaction. If q is the so-called charge of the domain element u, i.e., the coefficient of its singular part, then, in order to introduce a nonlinearity, we let the strength α depend on u according to the law α = −ν|q|σ, with ν > 0. This characterizes the model as a focusing NLS (nonlinear Schrodinger) with concentrated nonlinearity of power type. For such a ...

Wave equations with concentrated nonlinearities

In this paper, we address the problem of wave dynamics in the presence of concentrated nonlinearities. Given a vector field V on an open subset of and a discrete set with n elements, we define a nonlinear operator ΔV,Y on which coincides with the free Laplacian when restricted to regular functions vanishing at Y, and which reduces to the usual Laplacian with point interactions placed at Y when V is linear and represented by a Hermitian matrix. We then consider the nonlinear wave equation and study the corresponding Cauchy problem, giving an existence and uniqueness result when V is Lipschitz. The solution of such a problem is explicitly expressed in terms of the solutions of two Cauchy problems: one relative to a free wave equation and the other relative to an inhomogeneous ordinary differential equation with delay and principal part . The main properties of the solution are given and, when Y is a singleton, the mechanism and details of blow-up are studied.

The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

The nonlinear Schrödinger equation with spatially dependent nonlinearities has been considered in many contexts, both for its mathematical interest and for its relevance in several physical models. They represent situations in which there is a spatial inhomogeneity of the response of the medium to the wavefunction propagation. Here we are interested in the special case in which the nonlinearity is strongly concentrated in space around a finite number of points. More precisely, we investigate the limit where the nonlinearity becomes strictly pointlike. Such nonlinearities appear in several physical applications, for example: nonlinear diffraction of electrons from a thin layer (see [6]), analysis of nonlinear resonant tunneling (see [11]), models of pattern formation and bifurcation of solitons in nonlinear media (see [9, 12, 15, 16, 19] and references therein). Rigorous analysis of asymptotic stability of standing waves for some of the above models can be found in [7, 14]. To make precise the considered problem, we will consider the following nonlinear Schrödinger type equation

The point-like limit for a NLS equation with concentrated nonlinearity in dimension three

Abstract We consider a scaling limit of a nonlinear Schrodinger equation (NLS) with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a NLS equation with nonlinearity concentrated at a point. The regularized dynamics is described by the equation i ∂ ∂ t ψ e ( t ) = − Δ ψ e ( t ) + g ( e , μ , | ( ρ e , ψ e ( t ) ) | 2 μ ) ( ρ e , ψ e ( t ) ) ρ e where ρ e → δ 0 weakly and the function g embodies the nonlinearity and the scaling and has to be fine tuned in order to have a nontrivial limit dynamics. The limit dynamics is a nonlinear version of point interaction in dimension three and it has been previously studied in several papers as regards the well-posedness, blow-up and asymptotic properties of solutions. Our result is the first justification of the model as the point limit of a regularized dynamics.

Ground state and orbital stability for the NLS equation on a general starlike graph with potentials

We consider a nonlinear Schrodinger equation (NLS) posed on a graph or network composed of a generic compact part to which a finite number of half-lines are attached. We call this structure a starlike graph. At the vertices of the graph interactions of

On classical electrodynamics of point particles and mass renormalization: Some preliminary results

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