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...

We study dynamics of Dirac solitons in prototypical networks modeling them by the nonlinear Dirac equation on metric graphs. Stationary 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, conjectured to represent necessary conditions for allowing reflectionless transmission over a Y-junction are derived. The Y-junction considerations are also generalized to a tree network. The analytical results are confirmed by direct numerical simulations. PACS numbers:

Soliton transport in tubelike networks is studied by solving the nonlinear Schrödinger equation (NLSE) on finite thickness ("fat") graphs. The dependence of the solution and of the reflection at vertices on the graph thickness and on the angle between its bonds is studied and related to a special case considered in our previous work, in the limit when the thickness of the graph goes to zero. It is found that both the wave function and reflection coefficient reproduce the regime of reflectionless vertex transmission studied in our previous work.