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

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 Schrodinger 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.

Nonlinear Schr\"odinger 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 Schrodinger (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.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2013.0002

Nonlinear Schrödinger equation on graphs: recent results and open problems.

We investigate the existence of ground states for the subcritical NLS energy on metric graphs. In particular, we find out a topological assumption that guarantees the nonexistence of ground states, and give an example in which the assumption is not fulfilled and ground states actually exist. In order to obtain the result, we introduce a new rearrangement technique, adapted to the graph where it applies. Owing to such a technique, the energy level of the rearranged function is improved by conveniently mixing the symmetric and monotone rearrangement procedures.

NLS ground states on graphs

Threshold phenomena and existence results for NLS ground states on metric graphs

We study the case in which the nonlinear Schrodinger equation (NLSE) on simple networks consisting of vertices and bonds has an infinite number of constants of motion and becomes completely integrable just as in the case of a simple one-dimensional (1D) chain. Here the strength of cubic nonlinearity is different from bond to bond, and networks are assumed to have at least two semi-infinite bonds with one of them working as an incoming bond. The connection formula at vertices obtained from norm and energy conservation rules shows (1) the solution on each bond is a part of the universal (bond-independent) soliton solution of the completely integrable NLSE on the 1D chain, but is multiplied by the inverse of square root of bond-dependent nonlinearity; (2) nonlinearities at individual bonds around each vertex must satisfy a sum rule. Under these conditions, we also showed an infinite number of constants of motion. The argument on a branched chain or a primary star graph is generalized to other graphs, i.e., general star graphs, tree graphs, loop graphs and their combinations. As a relevant issue, with use of reflectionless propagation of Zakharov-Shabat's soliton through networks we have obtained the transmission probabilities on the outgoing bonds, which are inversely proportional to the bond-dependent strength of nonlinearity. Numerical evidence is also given to verify the prediction.

Integrable nonlinear Schrödinger equation on simple networks: Connection formula at vertices

We study dynamics of Dirac solitons in prototypical networks modeling them by the nonlinear Dirac equation on metric graphs. Soliton solutions of the nonlinear Dirac equation on simple metric graphs are obtained. It is shown that these solutions provide reflectionless vertex transmission of the Dirac solitons under suitable conditions. The constraints for bond nonlinearity coefficients, allowing reflectionless transmission over a Y-junction are derived. The analytical results are confirmed by direct numerical simulations.

/pdf/dynamics-of-dirac-solitons-in-networks-4gxdiodej8.pdf

Dynamics of Dirac solitons in networks

We consider branched quantum wires, whose connection rules provide PT-symmetry for the Schrodinger equation on graph. For such PT-symmetric quantum graph we derive general boundary conditions which keep the Hamiltonian as PT-symmetric with real eigenvalues and positively defined norm. Explicit boundary conditions which are consistent with the general PT-symmetric boundary conditions are presented. Secular equations for finding the eigenvalues of the quantum graph are derived. Breaking of the Kirchhoff rule at the branching point is shown. Experimental realization of PT-symmetric quantum graphs on branched optical waveguides is discussed.

PT-symmetric quantum graphs

/pdf/dynamics-of-dirac-solitons-in-networks-45t660cn6k.pdf

In this paper, we study quantum star graphs with time-dependent bond lengths. Quantum dynamics are treated by solving Schrodinger equation with time-dependent boundary conditions given on graphs. The time-dependence of the average kinetic energy is analyzed. The space-time evolution of a Gaussian wave packet is treated for an harmonically breathing star graph.

/pdf/time-dependent-quantum-graph-45pe6bedae.pdf

K. K. Sabirov

Papers

Integrable nonlinear Schrödinger equation on simple networks: Connection formula at vertices

Dynamics of Dirac solitons in networks

PT-symmetric quantum graphs

Dynamics of Dirac solitons in networks

Time-dependent quantum graph