Nonlinear Schrodinger equation and solitons on networks

Full text: The nonlinear Schrodinger equation (NLSE) is of importance in many areas of contemporary physics. The early applications of NLSE having soliton solutions were mainly focused 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-Pitaevskii 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. In this work we treat the stationary (cubic) nonlinear Schroedinger equation (NLSE) on simplest graphs. Motivation for the study of NLSE on graphs comes from the different practically important applications such as soliton transport in optical waveguide networks [40], solution dynamics in DNA double helix [1-3] and living systems [4] and other discrete structures [5]. An important applications of NLSE on networks is Bose-Einstein condensation (BEC) and transport of BEC in networks. The solutions are obtained for primary star graph with the boundary conditions providing vertex matching and flux conservation. Both, repulsive and attractive nonlinearities are considered. Explicit solutions of time-independent NLSE for primary star and tree graphs are obtained for matching and flux conservation boundary conditions. The method can be extended for other simplest topologies and their combinations. Unlike the previous studies [6,7], the lengths of the bonds are considered as finite. The results can be useful for the problems of BEC on networks and discrete traps, soliton transport optical waveguide networks, soliton excitation in DNA double helix, energy transfer in nanoscale networks etc. (authors) References: 1. S. Yomosa, Phys. Rev. A 27 , 2120 (1983). 2. C.T.Zhang, Phys. Rev. A 35 , 886 (1987). 3. L.V. Yakushevich, A.V. Savin, L.I. Manevitch, Phys. Rev. A 66 ,016614 (2002). 4. A.S. Davydov, Biology and Quantum Mechanics, (Oxford: Pergamon, 1982) 5. R. Burioni, D. Cassi, P. Sodano, A. Trombettoni, and A. Vezzani, Chaos 15, 043501 (2005); Physica D 216, 71 (2006). 6. Z. Sobirov, D. Matrasulov, K. Sabirov, S. Sawada, and K. Nakamura, Phys. Rev. E 81 , 066602 (2010). 7. R.Adami, C.Cacciapuoti, D.Finco, D.N., Rev.Math.Phys, 23 4 (2011).