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Stability of (1 + 1)-dimensional coupled nonlinear Schrödinger equation with elliptic potentials

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Abstract.

In this work, the (1 + 1)-dimensional coupled nonlinear Schrödinger equation with external elliptic potentials is studied. The exact expressions are obtained for spatially modulated periodic wave solutions in terms of Jacobi elliptic functions. The hyperbolic limits of these solutions are also considered. The cases of both attractive and repulsive nonlinear coefficients have been considered. The linear stability of these solutions is investigated both analytically and numerically. Different stable patterns, e.g., bright-bright and kink-kink solitons, are also obtained.

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References

  1. Y.S. Kivshar, G.P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003)

  2. P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-Gonzalez, Emergent Nonlinear Phenomena in Bose-Einstein Condensates (Springer-Verlag, Berlin, 2008)

  3. M.J. Ablowitz, B. Prinari, A.D. Trubatch, Discrete and Continuous Nonlinear Schrödinger Systems (Cambridge University Press, Cambridge, 2004)

  4. D.Y. Tang, N.R. Heckenberg, Phys. Rev. A 56, 1050 (1997)

    Article  ADS  Google Scholar 

  5. H. Pu, N.P. Bigelow, Phys. Rev. Lett. 80, 1130 (1998)

    Article  ADS  Google Scholar 

  6. T. Tsurumi, H. Morise, M. Wadati, Int. J. Mod. Phys. 14, 655 (2000)

    Article  ADS  Google Scholar 

  7. G. Theocharis, Z. Rapti, P.G. Kevrekidis, D.J. Frantzeskakis, V.V. Konotop, Phys. Rev. A 67, 063610 (2003)

    Article  ADS  Google Scholar 

  8. V. Achilleos, P.G. Kevrekidis, V.M. Rothos, D.J. Frantzeskakis, Phys. Rev. A 84, 053626 (2011)

    Article  ADS  Google Scholar 

  9. R. Mark Bradley, B. Deconinck, J.N. Kutz, J. Phys. A: Math. Gen. 38, 1901 (2005)

    Article  ADS  Google Scholar 

  10. T.S. Tsurumi, H. Morise, M. Wadati, Int. J. Mod. Phys. B 14, 655 (2000)

    Article  ADS  Google Scholar 

  11. B. Tan, J.P. Boyd, Chaos, Solitons Fractals 12, 721 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  12. T. Bartsch, Z.Q. Wang, J. Wei, J. Fixed Point Theory Appl. 2, 353 (2007)

    Article  MathSciNet  Google Scholar 

  13. N. Bottman, B. Deconinck, M. Nivala, J. Phys. A: Math. Theor. 44, 285201 (2011)

    Article  Google Scholar 

  14. O.C. Wright III, Physica D 264, 1 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  15. N. Nguyen, R. Tian, B. Deconinck, N.E. Sheils, J. Math. Phys. 54, 013701 (2013)

    Article  MathSciNet  Google Scholar 

  16. S.V. Manakov, Sov. Phys. JETP 38, 248 (1974)

    ADS  Google Scholar 

  17. M.G. Forest, D.W. McLaughlin, D.J. Muraki, O.C. Wright, J. Nonlinear Sci. 10, 291 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  18. J.C. Eilbeck, V.Z. Enolskii, N.A. Kostov, J. Math. Phys. 41, 8236 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  19. M.G. Forest, O.C. Wright, Physica D 178, 173 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  20. A.M. Kamchatnov, V.V. Sokolov, Phys. Rev. A 91, 043621 (2015)

    Article  ADS  Google Scholar 

  21. M.H. Jakubowski, K. Steiglitz, R. Squier, Phys. Rev. E 58, 6752 (1998)

    Article  ADS  Google Scholar 

  22. S.A. Derevyanko, J.E. Prilepsky, D.A Yakushev, J. Phys. A: Math. Gen. 39, 1297 (2006)

    Article  ADS  Google Scholar 

  23. B. Deconinck, J.N. Kutz, M.S. Patterson, B.W. Warner, J. Phys. A: Math. Gen. 36, 5431 (2003)

    Article  ADS  Google Scholar 

  24. N.A. Kostov, V.Z. Enol’skii, V.S. Gerdjikov, V.V. Konotop, M. Salerno, Phys. Rev. E 70, 056617 (2004)

    Article  ADS  Google Scholar 

  25. N.A. Kostov, V.S. Gerdjikov, T.I. Valchev, SIGMA 3, 071 (2007)

    Google Scholar 

  26. D. Yan, F. Tsitoura, P.G. Kevrekidis, D.J. Frantzeskakis, Phys. Rev. A 91, 023619 (2015)

    Article  ADS  Google Scholar 

  27. E.A. Ostrovskaya, Y.S. Kivshar, Phys. Rev. Lett. 92, 180405 (2004)

    Article  ADS  Google Scholar 

  28. M.A. Porter, P.G. Kevrekidis, B.A. Malomed, Physica D 196, 106 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  29. W. Hai, Y. Li, B. Xia, X. Luo, Europhys. Lett. 71, 28 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  30. M. Matuszewski, B.A. Malomed, M. Trippenbach, Phys. Rev. A 76, 043826 (2007)

    Article  ADS  Google Scholar 

  31. H.A. Cruz, V.A. Brazhnyi, V.V. Konotop, J. Phys. B: At. Mol. Opt. Phys. 41, 035304 (2008)

    Article  ADS  Google Scholar 

  32. R.M. Bradley, J.E. Bernard, L.D. Carr, Phys. Rev. A 77, 033622 (2008)

    Article  ADS  Google Scholar 

  33. P. Das, T.S. Raju, U. Roy, P.K. Panigrahi, Phys. Rev. 79, 015601 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  34. G.A. Sekh, M. Salerno, A. Saha, B. Talukdar, Phys. Rev. A 85, 023639 (2012)

    Article  ADS  Google Scholar 

  35. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (National Bureau of Standards, Washington, DC, 1964)

  36. D. Nath, B. Roy, R. Roychoudhury, Opt. Commun. 393, 224 (2017)

    Article  ADS  Google Scholar 

  37. J.C. Bronski, L.D. Carr, B. Deconinck, J.N. Kutz, Phys. Rev. Lett. 86, 1402 (2001)

    Article  ADS  Google Scholar 

  38. J.C. Bronski, L.D. Carr, B. Deconinck, J.N. Kutz, K. Promislow, Phys. Rev. E 63, 036612 (2001)

    Article  ADS  Google Scholar 

  39. L.D. Carr, J.N. Kutz, W.P. Reinhardt, Phys. Rev. E 63, 066604 (2001)

    Article  ADS  Google Scholar 

  40. J.C. Bronski, L.D. Carr, R. Carretero-González, B. Deconinck, J.N. Kutz, K. Promislow, Phys. Rev. E 64, 056615 (2001)

    Article  ADS  Google Scholar 

  41. B. Deconinck, B.A. Frigyik, J.N. Kutz, J. Nonlinear Sci. 12, 169 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  42. E. Kengne, R. Vaillancourt, B.A. Malomed, J. Phys. B: At. Mol. Opt. Phys. 41, 205202 (2008)

    Article  ADS  Google Scholar 

  43. B. Deconinck, N.E. Sheils, N.V. Nguyen, R. Tian, J. Phys. A: Math. Theor. 46, 415202 (2013)

    Article  Google Scholar 

  44. F.R. Gantmacher, Matrix Theory, Vol. 1 (AMS-Chelsea, Providence, Rhode Island, NY, 2000)

  45. R. Courant, D. Hilbert, Methods of Mathematical Physics (Wiley, New York, 1989)

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Authors and Affiliations

  1. Department of Mathematics, Vivekananda College, 700063, Kolkata, India

    Debraj Nath

  2. Physics & Applied Mathematics Unit, Indian Statistical Institute, 700108, Kolkata, India

    Naresh Saha & Barnana Roy

Authors
  1. Debraj Nath
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  2. Naresh Saha
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  3. Barnana Roy
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Correspondence to Debraj Nath.

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Nath, D., Saha, N. & Roy, B. Stability of (1 + 1)-dimensional coupled nonlinear Schrödinger equation with elliptic potentials. Eur. Phys. J. Plus 133, 504 (2018). https://doi.org/10.1140/epjp/i2018-12308-3

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  • DOI: https://doi.org/10.1140/epjp/i2018-12308-3

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