A variational principle for coupled nonlinear Schrödinger equations with variable coefficients and high nonlinearity. (English) Zbl 1219.65146

Zhou, Xin-Wei; Wang, Lin

Comput. Math. Appl. 61, No. 8, 2035-2038 (2011).

Summary: Via He’s semi-inverse method, a variational principle is established for coupled nonlinear Schrödinger equations with variable coefficients and high nonlinearity. The result includes previously known ones as special cases.

MSC:

65N99	Numerical methods for partial differential equations, boundary value problems
35Q55	NLS equations (nonlinear Schrödinger equations)
35A15	Variational methods applied to PDEs

Keywords:

variational principle; semi-inverse method; nonlinear Schrödinger equations

References:

[1]	Hasimoto, H.; Ono, H., Nonlinear modulation of gravity waves, J. Phys. Soc. Japan, 33, 805-811 (1972)
[2]	Johnson, R. S., On the modulation of water waves in the neighbourhood of \(k h \approx 1.363\), Proc. R. Soc. A, 357, 131-141 (1977) · Zbl 0375.76017
[3]	He, J. H., Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos Solitons Fractals, 19, 4, 847-851 (2004) · Zbl 1135.35303
[4]	Ambrosetti, A.; Colorado, E., Bound and ground states of coupled nonlinear Schrödinger equations, C. R. Math., 342, 7, 453-458 (2006) · Zbl 1094.35112
[5]	Xu, L., Variational principles for coupled nonlinear Schrödinger equations, Phys. Lett. A, 359, 627-629 (2007) · Zbl 1236.35175
[6]	Yao, L.; Chang, J. R., Variational principles for nonlinear Schrödinger equation with high nonlinearity, J. Nonlinear Sci. Appl., 1, 1, 1-4 (2008) · Zbl 1161.58307
[7]	Ozis, T.; Yildirim, A., Application of He’s semi-inverse method to the nonlinear Schrödinger equation, Comput. Math. Appl., 54, 7-8, 1039-1042 (2007) · Zbl 1157.65465
[8]	Sweilam, N. H.; Al-Bar, R. F., Variation iteration method for coupled nonlinear Schrödinger equation, Comput. Math. Appl., 54, 7-8, 993-999 (2007) · Zbl 1141.65387
[9]	Das, S.; Gupta, P. K.; Barat, S., A note on fractional Schrödinger equation, Nonlinear Sci. Lett. A, 1, 1, 91-94 (2010)
[10]	Kavitha, L.; Srividya, B.; Akila, N.; Gopi, D., Shape changing solitary solutions of a nonlocally damped nonlinear Schrödinger equation using symbolic computation, Nonlinear Sci. Lett. A, 1, 1, 95-107 (2010)
[11]	Yang, C. D.; Han, S. Y., Invariant eigen-structure in complex-valued quantum mechanics, Int. J. Nonlinear Sci. Numer., 10, 407-423 (2009)
[12]	Yang, C. D., Complex mechanics, Progr. Nonlinear Sci., 1, 1-383 (2010)
[13]	He, J. H., Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics, Int. J. Turbo Jet Eng., 14, 1, 23-28 (1997)
[14]	He, J. H., Variational approach to (2+1)-dimensional dispersive long water equations, Phys. Lett. A, 335, 2-3, 182-184 (2005) · Zbl 1123.37319
[15]	He, J. H., Variational principle for variable coefficients KdV equation, Phys. Lett. A, 358, 2, 91-93 (2006) · Zbl 1142.35583
[16]	Zhou, X. W., Variational approach to the Broer-Kaup-Kupershmidt equation, Phys. Lett. A, 363, 108-109 (2007) · Zbl 1197.65116
[17]	Zheng, C. B.; Liu, B.; Wang, Z. J., Generalized variational principle for electromagnetic field with magnetic monopoles by He’s semi-inverse method, Int. J. Nonlinear Sci. Numer., 10, 1369-1372 (2009)
[18]	Zheng, C. B.; Liu, B.; Wang, Z. J., Variational principle for nonlinear magneto-electro-elastodynamics with finite displacement by He’s semi-inverse method, Int. J. Nonlinear Sci. Numer., 10, 1523-1526 (2009)
[19]	He, J. H., Some asymptotic methods for strongly nonlinear equations, Internat. J. Modern Phys. B, 20, 10, 1141-1199 (2006) · Zbl 1102.34039
[21]	Wu, Y., Variational approach to the generalized Zakharov equations, Int. J. Nonlinear Sci. Numer., 10, 1245-1247 (2009)
[22]	Pak, S., Solitary wave solutions for the RLW equation by He’s semi inverse method, Int. J. Nonlinear Sci. Numer., 10, 505-508 (2009)
[23]	He, J. H.; Wu, X. H., Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, 30, 700-708 (2006) · Zbl 1141.35448
[24]	Wu, X. H. (Benn); He, J. H., Solitary solutions, periodic solutions and compacton-like solutions using the exp-function method, Comput. Math. Appl., 54, 7-8, 966-986 (2007) · Zbl 1143.35360
[25]	Zhou, X. W., Exp-function method for solving Huxley equation, Math. Probl. Eng., 2008 (2008), Article ID 538489 · Zbl 1151.92007
[26]	Zhou, X. W.; Wen, Y. X.; He, J. H., Exp-function method to solve the nonlinear dispersive \(K(m, n)\) equations, Int. J. Nonlinear Sci. Numer., 9, 3, 301-306 (2008)
[27]	Liu, W. J., New solitary wave solution for the Boussinesq wave equation using the semi-inverse method and the exp-function method, Z. Naturforsch. A, 64, 709-712 (2009)