Table of Contents Author Guidelines Submit a Manuscript
Differential Equations and Nonlinear Mechanics
Volume 2009, Article ID 395894, 29 pages
http://dx.doi.org/10.1155/2009/395894
Research Article

On Perturbative Cubic Nonlinear Schrodinger Equations under Complex Nonhomogeneities and Complex Initial Conditions

1Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Giza 12613, Egypt
2Department of Mathematics, Girls College, Medina, Saudi Arabia

Received 12 February 2009; Revised 30 May 2009; Accepted 8 July 2009

Academic Editor: Roger Grimshaw

Copyright © 2009 Magdy A. El-Tawil and Maha A. El-Hazmy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. J. Ablowitz, B. M. Herbst, and C. M. Schober, “The nonlinear Schrödinger equation: asymmetric perturbations, traveling waves and chaotic structures,” Mathematics and Computers in Simulation, vol. 43, no. 1, pp. 3–12, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. F. Kh. Abdullaev, J. C. Bronski, and G. Papanicolaou, “Soliton perturbations and the random Kepler problem,” Physica D, vol. 135, no. 3-4, pp. 369–386, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. S. Fewo, J. Atangana, A. Kenfack-Jiotsa, and T. C. Kofane, “Dispersion-managed solitons in the cubic complex Ginzburg-Landau equation as perturbations of nonlinear Schrodinger equation,” Optics Communications, vol. 252, pp. 138–149, 2005. View at Google Scholar
  4. A. Biswas and K. Porsezian, “Soliton perturbation theory for the modified nonlinear Schrödinger's equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 12, no. 6, pp. 886–903, 2007. View at Google Scholar · View at MathSciNet
  5. T. Cazenave and P.-L. Lions, “Orbital stability of standing waves for some nonlinear Schrödinger equations,” Communications in Mathematical Physics, vol. 85, no. 4, pp. 549–561, 1982. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A. Debussche and L. Di Menza, “Numerical simulation of focusing stochastic nonlinear Schrödinger equations,” Physica D, vol. 162, no. 3-4, pp. 131–154, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. Debussche and L. Di Menza, “Numerical resolution of stochastic focusing NLS equations,” Applied Mathematics Letters, vol. 15, no. 6, pp. 661–669, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. M. Wang, X. Li, and J. Zhang, “Various exact solutions of nonlinear Schrödinger equation with two nonlinear terms,” Chaos, Solitons and Fractals, vol. 31, no. 3, pp. 594–601, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. L.-P. Xu and J.-L. Zhang, “Exact solutions to two higher order nonlinear Schrödinger equations,” Chaos, Solitons and Fractals, vol. 31, no. 4, pp. 937–942, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. N. H. Sweilam, “Variational iteration method for solving cubic nonlinear Schrödinger equation,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 155–163, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S.-D. Zhu, “Exact solutions for the high-order dispersive cubic-quintic nonlinear Schrödinger equation by the extended hyperbolic auxiliary equation method,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1608–1612, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J.-Q. Sun, Z.-Q. Ma, W. Hua, and M.-Z. Qin, “New conservation schemes for the nonlinear Schrödinger equation,” Applied Mathematics and Computation, vol. 177, no. 1, pp. 446–451, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. K. Porsezian and B. Kalithasan, “Cnoidal and solitary wave solutions of the coupled higher order nonlinear Schrödinger equation in nonlinear optics,” Chaos, Solitons and Fractals, vol. 31, no. 1, pp. 188–196, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. H. Sakaguchi and T. Higashiuchi, “Two-dimensional dark soliton in the nonlinear Schrödinger equation,” Physics Letters A, vol. 359, no. 6, pp. 647–651, 2006. View at Google Scholar · View at MathSciNet
  15. D.-J. Huang, D.-S. Li, and H.-Q. Zhang, “Explicit and exact travelling wave solutions for the generalized derivative Schrödinger equation,” Chaos, Solitons and Fractals, vol. 31, no. 3, pp. 586–593, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. W. K. Abou Salem and C. Sulem, “Stochastic acceleration of solitons for the nonlinear Schrödinger equation,” SIAM Journal on Mathematical Analysis, vol. 41, no. 1, pp. 117–152, 2009. View at Google Scholar · View at MathSciNet
  17. M. El-Tawil, “The average solution of a stochastic nonlinear Schrodinger equation under stochastic complex non-homogeneity and complex initial conditions,” in Transactions on Computational Science III, vol. 5300 of Lecture Notes in Computer Science, pp. 143–170, Springer, New York, NY, USA, 2009. View at Google Scholar · View at Zentralblatt MATH
  18. M. Colin, T. Colin, and Mohta, “Stability of solitary eaves for a system of nonlinear Schrodinger equations with three wave interactions,” to appear in Annals de I'Institut Henri Poincare (c) Nonlinear Analysis.
  19. Z. Jia-Min and L. Yu-Lu, “Some exact solutions of variable coefficient cubic quintic nonlinear Schrodinger equation with an external potential,” Communications in Theoretical Physics, vol. 51, no. 3, p. 391, 2009. View at Google Scholar
  20. S. J. Farlow, Partial Differential Equations for Scientists and Engineers, John Wiley & Sons, New York, NY, USA, 1982. View at MathSciNet
  21. L. Pipes and L. Harvill, Applied Mathematics for Engineers and Physicists, McGraw-Hill, Tokyo, Japan, 1970.