zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Variational iteration method for coupled nonlinear Schrödinger equations. (English) Zbl 1141.65387
Summary: We apply the variational iteration method proposed by Ji-Huan He to simulate numerically a system of two coupled nonlinear one-dimensional Schrödinger equations subjected initially to a prescribed periodic wave solution. Test examples are given to demonstrate the accuracy and capability of the method with different wave-wave interaction coefficients. The accuracy of the method is verified by ensuring that the energy conservation remains almost constant. The numerical results obtained with a minimum amount of computation show that the variational iteration method is much easier, more convenient and efficient for solving nonlinear partial differential equations.

MSC:
65M70Spectral, collocation and related methods (IVP of PDE)
35Q55NLS-like (nonlinear Schrödinger) equations
WorldCat.org
Full Text: DOI
References:
[1] He, J. H.: Some asymptotic methods for strongly nonlinear equations. Internat. J. Modern phys. B 20, No. 10, 1141-1199 (2006) · Zbl 1102.34039
[2] He, J. H.: Variation iteration method -- A kind of non-linear analytical technique: some examples. Int. J. Non-linear mech. 34, 699-708 (1999) · Zbl 05137891
[3] Abdou, M. A.; Soliman, A. A.: New applications of variational iteration method. Physica D 211, No. 1--2, 1-8 (2005) · Zbl 1084.35539
[4] Abdou, M. A.; Soliman, A. A.: Variational iteration method for solving burger’s and coupled burger’s equations. J. comput. Appl. math. 181, No. 2, 245-251 (2005) · Zbl 1072.65127
[5] Abulwafa, E. M.; Abdou, M. A.; Mahmoud, A. A.: The solution of nonlinear coagulation problem with mass loss. Chaos solitons fractals 29, No. 2, 313-330 (2006) · Zbl 1101.82018
[6] Bildik, N.; Konuralp, A.: The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations. Int. J. Nonlinear sci. Numer. simul. 7, No. 1, 65-70 (2006) · Zbl 1115.65365
[7] Momani, S.; Abuasad, S.: Application of he’s varitional iteration method to Helmholtz equation. Chaos solitons fractals 27, 1119-1123 (2005) · Zbl 1086.65113
[8] Odibat, Z. M.; Momani, S.: Application of variational iteration method to nonlinear differential equations of fractional order. Int. J. Nonlinear sci. Numer. simul. 7, No. 1, 27-34 (2006) · Zbl 05675858
[9] N.H. Sweilam, M.M. Khader, Variational iteration method for one dimensional nonlinear thermoelasticity, Chaos Solitons Fractals (in press) · Zbl 1131.74018
[10] Soliman, A. A.: A numerical simulation and explicit solutions of KdV--Burgers and Lax’s seventh-order KdV equations. Chaos solitons fractals 29, No. 2, 294-302 (2006) · Zbl 1099.35521
[11] 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
[12] 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
[13] 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
[14] Tan, B.; Boyd, J. P.: Stability and long time evolution of the periodic solutions to two coupled nonlinear Schrödinger equations. Chaos solitons fractals 12, No. 4, 721-734 (2001) · Zbl 1022.35070
[15] Ablowitz, M. J.; Prinari, B.; Trubatch, A. D.: Discrete and continuous nonlinear Schrödinger systems. (2004) · Zbl 1057.35058
[16] Wright, O. C.: Modulational instability in a defocusing coupled nonlinear Schrödinger system. Physica D 82, 1-10 (1995) · Zbl 0900.35368
[17] Tan, B.; Boyd, J. P.: Coupled-mode envelope solitary waves in a pair of cubic Schrödinger equations with cross modulation: analytical solution and collisions with application to Rossby waves. Chaos solitons fractals 11, No. 7, 1113-1129 (2000) · Zbl 0945.35086
[18] Tan, B.; Liu, S.: Collision interactions of envelop Rossby solutions in a barotropic atmosphere. J. atoms sci. 53, 1604-1616 (1996)
[19] Tan, B.; Liu, S.: Collision interactions of solutions in a baroclinic atmosphere. J. atoms sci. 52, 1501-1512 (1995)
[20] Tsang, S. C.; Chow, K. W.: The evolution of periodic waves of the coupled nonlinear Schrödinger equations. Math. comput. Simul. 66, 551-564 (2004) · Zbl 1050.65078
[21] Zakhrov, V. E.; Schulman, E. I.: To the integrability of the system of two coupled nonlinear Schrödinger equations. Physica D 4, 270-274 (1982) · Zbl 1194.35435
[22] 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
[23] Fibich, G.; Gavish, N.; Wang, X.: New singular solutions of the nonlinear Schrödinger equation. Physica D 211, 193-220 (2005) · Zbl 1105.35114
[24] Forest, M. G.; Mclaughlin, D. W.; Muraki, D. J.; Wright, O. C.: Non-focusing instabilities in coupled, integrable, nonlinear Schrödinger pdes. J. nonlinear sci. 10, 291-331 (2000) · Zbl 0954.35145
[25] Forest, M. G.; Wright, O. C.: An integrable model for stable: unstable wave coupling phenomena. Physica D 178, 173-189 (2003) · Zbl 1011.74038
[26] Manakov, S. V.: On the theory of two-dimensional stationary self focusing of electromagnetic waves. Sov. phys. JETP 38, No. 2, 248-253 (1974)
[27] Rocha, J. A. E.: Numerical short-time behavior of Manakov system. Preprint
[28] Sun, J. Q.; Qin, M. Z.: Multi-symolectic methods for the coupled 1D nonlinear Schrödinger system. Comput. phys. Commun. 155, 221-235 (2003) · Zbl 1196.65195
[29] He, J. H.; Wu, X. H.: Construction of solitary solution and compacton-like solution by variational iteration method. Chaos solitons fractals 29, 108-113 (2006) · Zbl 1147.35338