×

A reliable treatment for nonlinear Schrödinger equations. (English) Zbl 1209.81104

Summary: Exp-function method is used to find a unified solution of nonlinear wave equation. Nonlinear Schrödinger equations with cubic and power law nonlinearity are selected to illustrate the effectiveness and simplicity of the method. It is shown that the Exp-function method, with the help of symbolic computation, provides a powerful mathematical tool for solving nonlinear equation.

MSC:

81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
68W30 Symbolic computation and algebraic computation
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] He, J.H., Int. J. non-linear. mech., 34, 4, 699, (1999)
[2] Darvishi, M.T.; Khani, F.; Soliman, A.A., Comput. math. appl., 54, 7-8, 1055, (2007)
[3] M.T. Darvishi, F. Khani, Numerical and explicit solutions of the fifth-order Korteweg – de Vries equations, Chaos Solitons Fractals (2007), doi: 10.1016/j.chaos.2007.07.034 · Zbl 1197.65165
[4] He, J.H., Int. J. mod. phys. B, 20, 18, 2561, (2006)
[5] He, J.H., Chaos solitons fractals, 26, 3, 695, (2005)
[6] He, J.H., Int. J. nonlinear sci. numer. simul., 6, 2, 207, (2005)
[7] A. Molabahrami, F. Khani, S. Hamedi-Nezad, Soliton solutions of the two-dimensional KdV-Burgers equation by homotopy perturbation method, Phys. Lett. A (2007), doi: 10.1016/j.physleta.2007.05.101 · Zbl 1209.65113
[8] A. Molabahrami, F. Khani, Numerical solutions of highly oscillatory integrals, Appl. Math. Comput. (2007), doi: 10.1016/j.amc.2007.09.007 · Zbl 1139.65019
[9] Wazwaz, A.M., Chaos solitons fractals, 25, 55, (2005)
[10] Abdusalam, H.A., Int. J. nonlinear sci. numer. simul., 6, 99, (2005)
[11] Bai, C.L.; Zhao, H., Chaos solitons fractals, 27, 1026, (2006)
[12] Abdou, M.A.; Soliman, A.A., Phys. lett. A, 353, 6, 487, (2006)
[13] Ibrahim, R.S.; El-Kalaawy, O.H., Chaos solitons fractals, 31, 4, 1001, (2007)
[14] El-Wakil, S.A.; Abdou, M.A., Chaos solitons fractals, 31, 4, 840, (2007)
[15] Franz, P.; Hongyou, W., J. geom. phys., 17, 3, 245, (1995)
[16] Wazwaz, A.M., Comput. math. appl., 50, 10-12, 1685, (2005)
[17] Xiqiang, Z.; Limin, W.; Weijun, S., Chaos solitons fractals, 28, 2, 448, (2006)
[18] Zhaosheng, F., Appl. math. comput., 158, 2, 593, (2004)
[19] Jie-Fang, Z., Phys. lett. A, 313, 5-6, 401, (2003)
[20] Fan, E.; Jian, Z., Phys. lett. A, 305, 6, 383, (2002)
[21] Fan, E.; Hon, Y.C., Appl. math. comput., 141, 351, (2003)
[22] Hon, Y.C.; Fan, E., Chaos solitons fractals, 24, 4, 1087, (2005)
[23] Yomba, E., Phys. lett. A, 340, 149, (2005)
[24] Yomba, E., Phys. lett. A, 336, 463, (2005)
[25] Yomba, E., Chaos solitons fractals, 26, 785, (2005)
[26] He, J.H., Int. J. mod. phys. B, 20, 10, 1141, (2006)
[27] He, J.H.; Wu, X.H., Chaos solitons fractals, 30, 3, 700, (2006)
[28] He, J.H.; Abdou, M.A., Chaos solitons fractals, 34, 1421, (2007)
[29] El-Wakil, S.A.; Madkour, M.A.; Abdou, M.A., Phys. lett. A, 369, 62, (2007)
[30] Zhang, S., Phys. lett. A, 365, 448, (2007)
[31] S. Zhang, Application of Exp-function method to high-dimensional nonlinear evolution equation, Chaos Solitons Fractals (2006), doi: 10.1016/j.chaos.2006.11.014
[32] Ebaid, A., Phys. lett. A, 365, 213, (2007)
[33] Ablowitz, M.; Segur, H., Solitons and the inverse scattering transform, (1981), SIAM Philadelphia · Zbl 0472.35002
[34] Drazin, P.G.; Johnson, R.S., Solitons: an introduction, (1993), Cambridge Univ. Press New York · Zbl 0661.35001
[35] Hirota, R., Direct methods in soliton theory, (1980), Springer Berlin
[36] Lax, P.D., Commun. pure appl. math., 28, 141, (1975)
[37] Wazwaz, A.M., Math. comput. mod., 43, 178, (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.