A solution of nonlinear Schrödinger equation on metric graphs
https://doi.org/10.17586/2220-8054-2015-6-2-162-172
Abstract
We treat the Nonlinear Schr¨odinger equation (NLSE) on Metric graph. An approach developed earlier for NLSE on interval [14], is extended for star graph. Dirichlet boundary conditions are imposed at the ends of bonds are imposed, while continuity conditions are chosen at the vertex of graph.
About the Authors
K. K. Sabirov
National University of Uzbekistan
Uzbekistan
Uzbekistan
4 University St., 100074, Tashkent
A. R. Khalmukhamedov
Tashkent branch of Moscow State University named after L. V. Lomonosov
Uzbekistan
Uzbekistan
22 Amir Temur St., 100060, Tashkent
References
1. Y.S. Kivshar and G.P. Agarwal. Optical Solitons: From Fibers to Photonic Crystals. Academic, San Diego (2003).
2. M.J. Ablowitz and P.A. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge: Cambridge University Press (1999).
3. C.J. Pethick and H. Smith. Bose-Einstein Condensation in Dilute Gases. Cambridge University Press, Cambridge, England (2002).
4. L. Pitaevskii and S. Stringari. Bose-Einstein Condensation. Oxford University Press, Oxford, England (2003).
5. T. Dauxois and M. Peyrard. Physics of Solitons. Cambridge University Press, Cambridge, England (2006).
6. Z. Sobirov, D. Matrasulov, K. Sabirov, et al. Integrable nonlinear Schrodinger equation on simple networks: Connection formula at vertices. Phys. Rev. E, 81, P. 066602 (2010).
7. R. Adami, C. Cacciapuoti, D. Finco, D. Noja. Fast solitons on star graphs. Reviews in Mathematical Physics, 23 (04), P. 409–451 (2011).
8. R. Adami, C. Cacciapuoti, D. Finco, D. Noja. On the structure of critical energy levels for the cubic focusing NLS on star graphs. Journal of Physics A: Mathematical and Theoretical, 45 (19), P. 192001 (2012).
9. R.C. Cascaval and C.T. Hunter. Linear and nonlinear Schrodinger equations on simple networks. Libertas Mathematica, 30, P. 85–98 (2010).
10. V. Banica and L.I. Ignat. Dispersion for the Schr¨odinger equation on networks Journal of Mathematical Physics; 52 (8), P. 083703–083703-14 (2011).
11. S. Gnutzmann, U. Smilansky, and S. Derevyanko. Stationary scattering from a nonlinear network. Phys. Rev. A, 83, P. 033831 (2011).
12. R. Adami, D. Noja, C. Ortoleva. Orbital and asymptotic stability for standing waves of a nonlinear Schr¨odinger equation with concentrated nonlinearity in dimension three. Journal of Mathematical Physics, 54, P. 013501–013533 (2013).
13. K.K. Sabirov, Z.A. Sobirov, D. Babajanov and D.U. Matrasulov. Stationary Nonlinear Schr¨odinger Equation on Simplest Graphs. Phys.Lett. A, 377, P. 860–865 (2013).
14. A.S. Fokas and A.R. Its. The nonlinear Schr¨odinger equation on the interval. J. Phys. A: Math. Gen, 37 P. 6091 (2004).
15. A.S. Fokas, A.R. Its and L.-Y. Sung. The nonlinear Schr¨odinger equation on the half-line. Nonlinearity, 18, P. 1771 (2005).
16. F. Harary. Graph Theory. Addison-Wesley, Reading (1969), 274 p.
17. T. Kottos and U. Smilansky. Periodic Orbit Theory and Spectral Statistics for Quantum Graphs. Ann.Phys., 274 (1) P. 76–124 (1999).
18. A. Boutet de Monvel and V.Kotlyarov. Scattering problem for the Zakharov-Shabat equations on the semi-axis. Inverse Problems, 16, P. 1813 (2000).
19. P.D. Lax. Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math., 21, P. 467–490 (1968).
20. V.E. Zakharov and A. Shabat. Exact theory of two-dimensional self-focusing and one-dimentional self-modulation of waves in nonlinear media. Sov. Phys. JETP, 34 (1), P. 62–69 (1972).
Review
For citations:
Sabirov K.K., Khalmukhamedov A.R. A solution of nonlinear Schrödinger equation on metric graphs. Nanosystems: Physics, Chemistry, Mathematics. 2015;6(2):162-172. https://doi.org/10.17586/2220-8054-2015-6-2-162-172
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