Time-Independent Nonlinear Schrödinger Equation on Simplest Networks

Conference paper
Part of the NATO Science for Peace and Security Series B: Physics and Biophysics book series (NAPSB)


We treat the time-independent (cubic) nonlinear Schrödinger equation (NLSE) on simplest networks. In particular, the solutions are obtained for star and tree graphs with the boundary conditions providing vertex matching and flux conservation. It is shown that the method can be extended to the case of arbitrary number of bonds in star graphs and for other simplest topologies.


Soliton Solution Simple Graph Nonlinear Evolution Equation Schrodinger Equation Tree Graph 
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  1. 1.
    Zakharov VE, Shabat AB (1972) Sov Phys JETP 34:62MathSciNetADSGoogle Scholar
  2. 2.
    Zakharov VE, Shabat AB (1974) Funct Anal Appl 8:226zbMATHCrossRefGoogle Scholar
  3. 3.
    Zakharov VE, Shabat AB (1979) Funct Anal Appl 13:166MathSciNetGoogle Scholar
  4. 4.
    Kivshar YS, Agarwal GP (2003) Optical solitons: from fibers to photonic crystals. Academic, San DiegoGoogle Scholar
  5. 5.
    Ablowitz MJ, Clarkson PA (1999) Solitons, nonlinear evolution equations and inverse scattering. Cambridge University Press, CambridgeGoogle Scholar
  6. 6.
    Pethick CJ, Smith H (2002) Bose-Einstein condensation in dilute gases. Cambridge University Press, CambridgeGoogle Scholar
  7. 7.
    Pitaesvki L, Stringari S (2003) Bose-Einstein condensation. Oxford University Press, OxfordGoogle Scholar
  8. 8.
    Dauxois T, Peyrard M (2006) Physics of solitons. Cambridge University Press, CambridgezbMATHGoogle Scholar
  9. 9.
    Carr LD, Clark CW, Reinhardt WP (2000) Phys Rev A 62:063610ADSCrossRefGoogle Scholar
  10. 10.
    Carr LD, Clark CW, Reinhardt WP (2000) Phys Rev A 62:063611ADSCrossRefGoogle Scholar
  11. 11.
    DAgosta R, Malomed BA, Presilla C (2000) Phys Lett A 275:424Google Scholar
  12. 12.
    Carr LD, Mahmud KW, Reinhardt WP (2001) Phys Rev A 64:033603ADSCrossRefGoogle Scholar
  13. 13.
    Rapedius K, Witthaut D, Korsch HJ (2006) Phys Rev A 73:033608ADSCrossRefGoogle Scholar
  14. 14.
    Infeld E, Zin P, Gocalek J, Trippenbach M (2006) Phys Rev E 74:026610ADSCrossRefGoogle Scholar
  15. 15.
    Harary F (1969) Graph theory. Addison-Wesley, ReadingGoogle Scholar
  16. 16.
    Kottos T, Smilansky U (1999) Ann Phys 76:274MathSciNetGoogle Scholar
  17. 17.
    Gnutzmann S, Smilansky U (2006) Adv Phys 55:527ADSCrossRefGoogle Scholar
  18. 18.
    Gnutzmann S, Keating JP, Piotet F (2010) Ann Phys 325:2595MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Exner P, Seba P, Stovicek P (1988) J Phys A 21:4009–4019MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    Exner P, Seba P (1989) Rep Math Phys 28:7MathSciNetADSzbMATHCrossRefGoogle Scholar
  21. 21.
    Kostrykin V, Schrader R (1999) J Phys A 32:595MathSciNetADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Kuchment P (2004) Wave Random Media 14:S107MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Burioni R, Cassi D, Sodano P, Trombettoni A, Vezzani A (2006) Chaos 15:043501; Physica D 216:71Google Scholar
  24. 24.
    Leboeuf P, Pavloff N (2001) Phys Rev A 64:033602ADSCrossRefGoogle Scholar
  25. 25.
    Bongs K et al (2001) Phys Rev A 63:031602 (R)Google Scholar
  26. 26.
    Paul T, Hartung M, Richter K, Schlagheck P (2007) Phys Rev A 76:063605ADSCrossRefGoogle Scholar
  27. 27.
    de Oliveira IN (2010) Phys Rev E 81:030104(R)Google Scholar
  28. 28.
    Sobirov Z, Matrasulov D, Sabirov K, Sawada S, Nakamura K (2010) Phys Rev E 81:066602MathSciNetADSCrossRefGoogle Scholar
  29. 29.
    Adami R, Cacciapuoti C, Finco D, Noja D (2011) Rev Math Phys 23:4MathSciNetCrossRefGoogle Scholar
  30. 30.
    Cascaval RC, Hunter CT (2010) Libertas Math 30:85MathSciNetzbMATHGoogle Scholar
  31. 31.
    Banica V, Ignat L (2011) J Phys A 52:083703MathSciNetGoogle Scholar
  32. 32.
    Gnutzmann S, Smilansky U, Derevyanko S (2011) Phys Rev A 83:033831ADSCrossRefGoogle Scholar
  33. 33.
    Adami R, Cacciapuoti C, Finco D, Noja D (2012) J Phys A 45:192001MathSciNetADSCrossRefGoogle Scholar
  34. 34.
    Bowman F (1961) Introduction to elliptic functions, with applications. Dover, New YorkzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of PhysicsNational University of UzbekistanTashkentUzbekistan
  2. 2.Department of EnergyTurin Polytechnic University in TashkentTashkentUzbekistan
  3. 3.Department of Energy, Laboratory for Advanced StudiesTurin Polytechnic University in TashkentTashkentUzbekistan

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