We study the case in which the nonlinear Schrödinger 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.

This is the published version of the paper.This version of the publication may differ from the final published version. We study electric dipole effects for massive Dirac fermions in graphene and related materials. The dipole potential accommodates towers of infinitely many bound states exhibiting a universal Efimov-like scaling hierarchy. The dipole moment determines the number of towers, but there is always at least one tower. The corresponding eigenstates show a characteristic angular asymmetry, observable in tunnel spectroscopy. However, charge transport properties inferred from scattering states are highly isotropic. Permanent

We consider the sine-Gordon equation on metric graphs with simple topologies and derive vertex boundary conditions from the fundamental conservation laws together with successive spacederivatives of sine-Gordon equation. We analytically obtain traveling wave solutions in the form of standard sine-Gordon solitons such as kinks and antikinks for star and tree graphs. We show that for this case the sine-Gordon equation becomes completely integrable just as in case of a simple 1D chain. This simple analysis provides a cornerstone for the numerical solution of the general case, including a quantification of the vertex scattering. Applications of the obtained results to Josephson junction networks and DNA double helix are discussed.

We treat the stationary (cubic) nonlinear Schrödinger equation (NSLE) on simplest graphs. Formulation of the problem and exact analytical solutions of NLSE are presented for star graphs consisting of three bonds. It is shown that the method can be extended for the case of arbitrary number of bonds of star graphs and for other simplest topologies such as tree and loop graphs. The case of repulsive and attractive nonlinearities are treated separately. : 05.45.Yv, 42.65.Tg, 42.65.Wi, 05.60.Gg. PACS I. INTRODUCTION.The nonlinear Schrödinger equation has attracted much attention since from its pioneering studies in early seventies of the last century [1]- [3]. Such attention was caused by the possibility for obtaining soliton solution of NLSE and its numerous applications in different branches of physics. The early applications of NLSE and other nonlinear PDEs having soliton solutions were mainly focussed in optics, acoustics, particle physics, hydrodynamics and biophysics. However, special attention NLSE and its soliton solutions have attracted because of the recent progress made in the physics and Bose-Einstein condensates(BEC). Namely, due to the fact that the dynamics of BEC is governed by Gross-Pitaevski equation which is NLSE with cubic nonlinearity, finding the soliton solution of NLSE with different confining potentials and boundary conditions is of importance for this area of physics.Many aspects of soliton solution of NLSE have been treated during the past decade in the context of fiber optics, photonic crystals, acoustics and BEC (see books [4]-[8] and references therein). Both, stationary and time-dependent NLSE were extensively studied for different trapping potentials in the context of BEC. In particular, the stationary NLSE was studied for box boundary conditions [9, 10] and the square well potential [11]- [14].In this paper we treat the stationary NLSE on graphs. Graphs are the systems consisting of bonds which are connected at the vertices [15]. The bonds are connected according to a rule that is called topology of a graph. Topology of a graph is given in terms of so-called adjacency matrix (or connectivity matrix) which can be written as [16,17]:1 if i and j are connected, 0 otherwise, i, j = 1, 2, ..., V.The linear Schrödinger equation on graphs has been topic of extensive research recently (e.g., see review [16]-[18] and references therein). In this case the eigenvalue problem is given in terms of the boundary conditions providing continuity and current conservation [16]-[20].Despite the progress made in the study of linear Schrödinger equation on graphs, corresponding nonlinear problem, i.e., NLSE on graphs is still remaining as less-studied problem. This is mainly caused the difficulties that appear in the case of NLSE on graphs, especially, for the time-dependent problem. In particular, the problem becomes rather nontrivial and it is not so easy to derive conservation laws [33]. It should be noted that during the last couple of years there were some attempts to treat time-dependent [33,34] and t...