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The Discrete Nonlinear Schrödinger Equation pp 221–233Cite as

A Map Approach to Stationary Solutions of the DNLS Equation

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Part of the book series: Springer Tracts in Modern Physics ((STMP,volume 232))

In this chapter we discuss the well-established map approach for obtaining stationary solutions to the one-dimensional (1D) discrete nonlinear Schrödinger (DNLS) equation. The method relies on casting the ensuing stationary problem in the form of a recurrence relationship that can in turn be cast into a two-dimensional (2D) map [1–5]. Within this description, any orbit for this 2D map will correspond to a steady state solution of the original DNLS equation.

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References

  1. Hennig, D., Rasmussen, K.Ø., Gabriel, H., Bülow, A.: Phys. Rev. E 54, 5788 (1996)

    Google Scholar 

  2. Bountis, T., Capel, H.W., Kollmann, M., Ross, J.C., Bergamin, J.M., van der Weele, J.P.: Phys. Lett. A 268, 50 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Alfimov, G.L., Brazhnyi, V.A., Konotop, V.V.: Physica D 194, 127 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  4. Carretero-Gonzáles, R., Talley, J.D., Chong, C., Malomed, B.A.: Physica D 216, 77 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  5. Qin, W.X., Xiao, X.: Nonlinearity 20 2305 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  6. Maluckov, A., Hadžievski, L., Malomed, B.A.: Phys. Rev. E 77, 036604 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  7. Hennig, D., Tsironis, G.P.: Phys. Rep. 307 333 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  8. Malomed, B.A., Kevrekidis, P.G.: Phys. Rev. E 64, 026601 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  9. Kevrekidis, P.G., Malomed, B.A., Chen, Z., Frantzeskakis, D.J.: Phys. Rev. E 70, 056612 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  10. Sakaguchi, H., Malomed, B.A.: Europhys. Lett. 72, 698 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  11. Neshev, D.N., Alexander, T.J., Ostrovskaya, E.A., Kivshar, Yu.S., Martin, H., Makasyuk, I., Chen, Z.: Phys. Rev. Lett. 92, 123903 (2004)

    Article  ADS  Google Scholar 

  12. Fleischer, J.W., Bartal, G., Cohen, O., Manela, O., Segev, M., Hudock, J., Christodoulides, D.N.: Phys. Rev. Lett. 92, 123904 (2004)

    Article  ADS  Google Scholar 

  13. Kevrekidis, P.G., Malomed, B.A., Frantzeskakis, D.J., Carretero-González, R.: Phys. Rev. Lett. 93, 080403 (2004)

    Article  ADS  Google Scholar 

  14. Carretero-González, R., Kevrekidis, P.G., Malomed, B.A., Frantzeskakis, D.J.: Phys. Rev. Lett. 94, 203901 (2005)

    Article  ADS  Google Scholar 

  15. Kevrekidis, P.G., Carretero-González, R., Frantzeskakis, D.J., Malomed, B.A., Diakonos, F.K.: Phys. Rev. E 75, 026603 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  16. Balasuriya, S.: Physica D, 202, 155 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  17. Hennig, D., Sun, N.G., Gabriel, H., Tsironis, P.: Phys. Rev. E 52, 255 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  18. Glasser, M.L., Papageorgiou, V.G., Bountis, T.C.: SIAM J. Appl. Math. 49, 692 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kollmann, M., Bountis, T.: Physica D 113, 397 (1998)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Bountis, T., Goriely, A., Kollmann, M.: Phys. Lett. A 206, 38 (1995)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. James, G., Sánchez-Rey, B., Cuevas, J.: Preprint. arXiv:nlin.PS/0710.4114 154

    Google Scholar 

  22. Cuevas, J., James, G., Malomed, B.A., Kevrekidis, P.G., Sánchez-Rey, B.: Preprint (2007). ArXiv:nlin.PS/0712.1837 154

    Google Scholar 

  23. Palmero, F., Carretero-González, R., Cuevas, J., Kevrekidis, P.G., Królikowski, W.: Phys. Rev. E 77 036614 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  24. Carretero-González, R., Promislow, K.: Phys. Rev. A 66, 033610 (2002).

    Article  ADS  Google Scholar 

  25. Susanto, H., Johansson, M.: Phys. Rev. E 72, 016605 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  26. Pelinovsky, D.E., Kevrekidis, P.G.: J. Phys. A: Math. Theor. 41 185206 (2008)

    Article  MathSciNet  Google Scholar 

  27. Debnath, L.: Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhäuser Boston, (2005)

    Google Scholar 

  28. MacKay, R.S., Aubry, S.: Nonlinearity 7, 1623 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. Dullin, H.R., Meiss, J.D.: Physica D 143, 265 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  30. Sterling, D., Dullin, H.R., Meiss, J.D.: Physica D 134, 153 (1999)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  31. Bak, P.: Phys. Today 12 38 (1986)

    Article  Google Scholar 

  32. Schroeder, M.: Fractals, Chaos, Power Laws. W.H. Freeman and Company, 1991. Chap. 7

    Google Scholar 

  33. Carretero-González, R., Arrowsmith, D.K., Vivaldi, F.: Physica D 103 381 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Palmer, J.K.: J. Diff. Eqns. 55, 225 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  35. Chow, S.N., Hale, J.K., Mallet-Paret, J.: J. Diff. Eqns. 37, 351 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  36. Chow, S.N., Yamashita, M.: Nonlinear Equations in the Applied Sciences. Academic, New York, (1992)

    Google Scholar 

  37. Ablowitz, M.J., Ladik, J.F.: J. Math. Phys. 16, 598 (1975)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Ablowitz, M.J., Ladik, J.F.: J. Math. Phys. 17, 1011 (1976)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. Bülow, A., Hennig, D., Gabriel, H.: Phys. Rev. E 59, 2380 (1999)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Nonlinear Dynamical Systems Group, Computational Science Research Center, and Department of Mathematics and Statistics, San Diego State University, San Diego, CA, 92182-7720, USA

    Ricardo Carretero-González

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  1. Ricardo Carretero-González
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Correspondence to Ricardo Carretero-González .

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Carretero-González, R. (2009). A Map Approach to Stationary Solutions of the DNLS Equation. In: The Discrete Nonlinear Schrödinger Equation. Springer Tracts in Modern Physics, vol 232. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-89199-4_11

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  • DOI: https://doi.org/10.1007/978-3-540-89199-4_11

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