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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.

In the present note, we study the focusing NLS equation in dimension two with a point interaction in the supercritical regime, showing two results. After obtaining the (nonstandard) virial formula, we exhibit a set of initial data that shows blow-up. Moreover, we show that the standing waves e i ω t φ ω corresponding to ground states φ ω of the action functional are strongly unstable, at least for sufficiently high ω .

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.

We study standing waves of the NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices, Kirchhoff boundary conditions are imposed. We pursue a recent study concerning solutions nonzero on the half-lines and periodic on the circle, by proving some existing results of sign-changing solutions non-periodic on the circle.

We consider Schr\"odinger operators on a bounded domain \\(\\Omega\\subset \\mathbb{R}^3\\), with homogeneous Robin or Dirichlet boundary conditions on \\(\\partial\\Omega\\) and a point (zero-range) interaction placed at an interior point of \\(\\Omega\\). We show that, under suitable spectral assumptions, and by means of an extension-restriction procedure which exploit the already known result on the entire space, the singular interaction is approximated by rescaled sequences of regular potentials. The result is missing in the literature, and we also take the opportunity to point out some general issues in the approximation of point interactions and the role of zero energy resonances.

In this paper we describe the resonances of the singular perturbation of the Laplacian on the half space \\(\\Omega =\\mathbb R^3_+\\) given by the self-adjoint operator named \\(\\delta\\)-interaction or point interaction. We will assume Dirichlet or Neumann boundary conditions on \\(\\partial \\Omega\\). At variance with the well known case of \\(\\mathbb R^3\\), the resonances constitute an infinite set, here completely characterized. Moreover, we prove that resonances have an asymptotic distribution satisfying a modified Weyl law. Finally we give applications of the results to the asymptotic behavior of the abstract Wave and Schr\"odinger dynamics generated by the Laplacian with a point interaction on the half-space.

We review evolutionary models on quantum graphs expressed by linear and nonlinear partial differential equations. Existence and stability of the standing waves trapped on quantum graphs are studied by using methods of the variational theory, dynamical systems on a phase plane, and the Dirichlet-to-Neumann mappings.

We study the Korteweg--de Vries equation on a metric star graph and investigate existence of solitary waves on the metric graph in terms of the coefficients of the equation on each edge, the coupling condition at the central vertex of the star and the speeds of the travelling wave. We show that, with a continuity condition at the vertex, solitary waves can occur exactly when the parameters are chosen in a fairly special manner. We also consider coupling conditions beyond continuity.

In the present 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 nonlinear Schr\"odinger equation. Special consideration is given to existence and behaviour of solitary solutions. Two subjects are discussed in some detail concerning NLS equation on a star graph: the standing waves of NLS equation on a graph with a \\(\\delta\\) interaction at the vertex; the scattering of fast solitons through an Y-junction in the cubic case. The emphasis is on description of concepts and results and on physical context, without reporting detailed proofs; some perspectives and more ambitious open problems are discussed.

The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schr\"{o}dinger equation with quintic power nonlinearity equipped with the Neumann-Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency \\(\\omega\\in (-\\infty,0)\\) is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in \\(L^6\\). The set of minimizers includes the set of ground states of the system, which are the global minimizers of the energy at constant mass (\\(L^2\\)-norm), but it is actually wider. While ground states exist only for a certain interval of masses, the standing waves exist for every \\(\\omega \\in (-\\infty,0)\\) and correspond to a bigger interval of masses. It is shown that there exist critical frequencies \\(\\omega_0\\) and \\(\\omega_1\\) such that the standing waves are the ground states for \\(\\omega \\in [\\omega_0,0)\\), local minimizers of the energy at constant mass for \\(\\omega \\in (\\omega_1,\\omega_0)\\), and saddle points of the energy at constant mass for \\(\\omega \\in (-\\infty,\\omega_1)\\). Proofs make use of both the variational methods and the analytical theory for differential equations.