A study on linear and nonlinear Schrödinger equations by the variational iteration method. (English) Zbl 1148.35353

Summary: We introduce a framework to obtain exact solutions to linear and nonlinear Schrödinger equations. The He’s variational iteration method (VIM) is used for analytic treatment of these equations. Numerical examples are tested to show the pertinent features of this method.


35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs
Full Text: DOI


[1] He, J. H., Some asymptotic methods for strongly nonlinear equations, Int J Mod Phys B, 20, 10, 1141-1199 (2006) · Zbl 1102.34039
[3] He, J. H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput Meth Appl Mech Eng, 167, 57-68 (1998) · Zbl 0942.76077
[4] He, J. H., Variational iteration method for autonomous ordinary differential systems, Appl Math Comput, 114, 2/3, 115-123 (2000) · Zbl 1027.34009
[5] He, J. H., Homotopy perturbation method: a new nonlinear technique, Appl Math Comput, 135, 73-79 (2003) · Zbl 1030.34013
[6] He, J. H., A new approach to nonlinear partial differential equations, Comm Nonlinear Sci Numer Simul, 2, 4, 203-205 (1997)
[7] He, J. H., A variational iteration approach to nonlinear problems and its applications, Mech Appl, 20, 1, 30-31 (1998)
[8] He, J. H., Variational iteration method-a kind of nonlinear analytical technique: some examples, Int J Nonlinear Mech, 34, 708-799 (1999)
[9] He, J. H., A generalized variational principle in micromorphic thermoelasticity, Mech Res Commun, 3291, 93-98 (2005) · Zbl 1091.74012
[10] He, J. H., The homotopy perturbation method for nonlinear oscillators with discontinuities, Appl Math Comput, 151, 287-292 (2004) · Zbl 1039.65052
[11] Abulwafa, E. M.; Abdou, M. A.; Mahmoud, A. A., The solution of nonlinear coagulation problem with mass loss, Chaos, Solitons & Fractals, 29, 313-330 (2006) · Zbl 1101.82018
[12] Momani, S.; Odibat, Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys Lett A, 1, 53, 1-9 (2006)
[13] Momani, S.; Abusaad, S., Application of He’s variational-iteration method to Helmholtz equation, Chaos, Solitons & Fractals, 27, 5, 1119-1123 (2005) · Zbl 1086.65113
[14] Abdou, M. A.; Soliman, A. A., variational iteration method for solving Burgers’ and coupled Burgers’ equation, J Comput Appl Math, 181, 245-251 (2005) · Zbl 1072.65127
[17] Wazwaz, A. M., Partial differential equations: methods and applications (2002), Balkema Publishers: Balkema Publishers The Netherlands · Zbl 0997.35083
[18] Wazwaz, A. M., Necessary conditions for the appearance of noise terms in decomposition solution series, Appl Math Comput, 81, 265-274 (1997) · Zbl 0882.65132
[19] Wazwaz, A. M., A new technique for calculating Adomian polynomials for nonlinear polynomials, Appl Math Comput, 111, 1, 33-51 (2000)
[20] Wazwaz, A. M., A new method for solving singular initial value problems in the second order differential equations, Appl Math Comput, 128, 47-57 (2002) · Zbl 1030.34004
[21] Wazwaz, A. M., A first course in integral equations (1997), World Scientific: World Scientific Singapore
[22] Wazwaz, A. M., Analytical approximations and Padé’ approximants for Volterra’s population model, Appl Math Comput, 100, 13-25 (1999) · Zbl 0953.92026
[23] Adomian, G., Solving frontier problems of physics: the decomposition method (1994), Kluwer: Kluwer Boston · Zbl 0802.65122
[24] Adomian, G., A review of the decomposition method in applied mathematics, J Math Anal Appl, 135, 501-544 (1988) · Zbl 0671.34053
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.