Variational iteration method for solving cubic nonlinear Schrödinger equation.(English)Zbl 1119.65098

Summary: The variational iteration method is applied to solve the cubic nonlinear Schrödinger (CNLS) equation in one and two space variables. In both cases, we will reduce the CNLS equation to a coupled system of nonlinear equations. Numerical experiments are made to verify the efficiency of the method. Comparison with the theoretical solution shows that the variational iteration method is of high accuracy.

MSC:

 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text:

References:

 [1] M.A. Abdou, A.A. Soliman, Variational iteration method for solving Burger’s and coupled Burger’s equations, J. Comput. Appl. Math. (2004) 1-7. [2] Abdou, M.A.; Soliman, A.A., New applications of variational iteration method, Physica D, 211, 1-8, (2005) · Zbl 1084.35539 [3] Ablowitz, M.J.; Prinari, B.; Trubatch, A.D., Discrete and continuous nonlinear Schrödinger systems, Bulletin (new series) of the amer. math. soc., 43, 1, 127-132, (2005) [4] Argyris, J.; Haase, M., An Engineer’s guide to solitons phenomena: application of the finite element method, Comput. methods appl. mech. eng., 61, 71-122, (1987) · Zbl 0624.76020 [5] Carter, J.D.; Deconinck, B., Instabilities of one-dimensional trivial-phase solutions of the two-dimensional cubic nonlinear Schrödinger equation, Physica D, 214, 42-54, (2006) · Zbl 1091.35086 [6] Drăgănnescu, Gh.E.; Căpălnăsan, V., Nonlinear relaxation phenomena in polycrystalline solids, Internat. J. nonlinear sci. numer. simulation, 4, 219-225, (2003) [7] Drăgănnescu, Gh.E.; Cofan, N.; Rujan, D.L., Nonlinear vibrations a nano-sized sensor with fractional damping, J. optoelectronics adv. mater., 7, 2, 877-884, (2005) [8] El-Sayed, S.M.; Kaya, D., On the numerical solution of the system of two-dimensional Burger’s equations by the decomposition method, Appl. math. comput., 158, 101-109, (2004) · Zbl 1061.65099 [9] Th. Famelis, A.G. Bratsos, A solution of the cubic Schrödinger equation using the Adomian decomposition method, preprint, available in $$\langle$$www.aueb.gr/pympe/hercma/proceedings2005/H05-ABSTRACTS-1/PAPADOPOULOS-FAMELIS-BRATSOS-1.pdf⟩. [10] Fibich, G.; Gavish, N.; Wang, X., New singular solutions of the nonlinear Schrödinger equation, Physica D, 211, 193-220, (2005) · Zbl 1105.35114 [11] He, J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. methods appl. mech. eng., 167, 57-68, (1998) · Zbl 0942.76077 [12] He, J.H., Approximate solution of nonlinear differential equations with convolution product nonlinearities, Comput. methods appl. mech. eng., 167, 69-73, (1998) · Zbl 0932.65143 [13] He, J.H., Variational iteration method—a kind of non-linear analytical technique: some examples, Internat. J. non-linear mech., 34, 699-708, (1999) · Zbl 1342.34005 [14] He, J.H., Variational iteration method for autonomous ordinary differential systems, Appl. math. comput., 114, 115-123, (2000) · Zbl 1027.34009 [15] He, J.H., Some asymptotic methods for strongly nonlinear equations, Internat. J. modern phys. B, 20, 10, 1141-1199, (2006) · Zbl 1102.34039 [16] J.H. He, Non-perturbative methods for strongly nonlinear problems, Berlin: dissertation.de-Verlag im Internet GmbH, 2006. [17] He, J.H.; Wan, Y.Q.; Guo, Q., An iteration formulation for normalized diode characteristics, Internat. J. circuit theory appl., 32, 629-632, (2004) · Zbl 1169.94352 [18] He, J.H.; Wu, X.H., Construction of solitary solution and compaction-like solution by variational iteration method, Chaos solitons fractals, 29, 1, 108-113, (2006) · Zbl 1147.35338 [19] Kaya, D.; El-Sayed, S.M., On the solution of the couples schrödinger – kdv equation by the decomposition method, Phys. lett. A, 313, 82-88, (2003) · Zbl 1040.35099 [20] Momani, S.; Abuasad, S., Application of He’s variational iteration method to Helmholtz equation, Chaos solitons fractals, 27, 1119-1123, (2005) · Zbl 1086.65113 [21] Sanz-Serna, J.M., Conservative and nonconservative schemes for the solution of non-linear Schrödinger equation, IMA J. numer. anal., 6, 25-42, (1986) · Zbl 0593.65087 [22] N.H. Sweilam, M.M. Khader, Variational iteration method for one dimensional nonlinear thermoelasticity, Chaos Solitons Fractals (2005). · Zbl 1131.74018 [23] 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 [24] Wazwaz, A.M., A reliable algorithm for obtaining positive solutions for nonlinear boundary value problems, Comput. math. appl., 41, 1237-1244, (2001) · Zbl 0983.65090
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.