×

Generalized hyperbolic function solution to a class of nonlinear Schrödinger-type equations. (English) Zbl 1246.35186

Summary: With the help of the generalized hyperbolic function, the subsidiary ordinary differential equation method is improved and proposed to construct exact traveling wave solutions of the nonlinear partial differential equations in a unified way. A class of nonlinear Schrödinger-type equations including the generalized Zakharov system, the Rangwala-Rao equation, and the Chen-Lee-Liu equation are investigated and the exact solutions are derived with the aid of the homogenous balance principle and generalized hyperbolic functions. We study the effect of the generalized hyperbolic function parameters \(p\) and \(q\) in the obtained solutions by using computer simulations.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35C07 Traveling wave solutions
35A24 Methods of ordinary differential equations applied to PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0900.65350
[2] E. Fan and Y. C. Hon, “Generalized tanh method extended to special types of nonlinear equations,” Zeitschrift fur Naturforschung A, vol. 57, no. 8, pp. 692-700, 2002.
[3] A. H. Bokhari, G. Mohammad, M. T. Mustafa, and F. D. Zaman, “Adomian decomposition method for a nonlinear heat equation with temperature dependent thermal properties,” Mathematical Problems in Engineering, vol. 2009, Article ID 926086, 12 pages, 2009. · Zbl 1181.80002
[4] A.-M. Wazwaz, “The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations,” Applied Mathematics and Computation, vol. 167, no. 2, pp. 1196-1210, 2005. · Zbl 1082.65585
[5] J.-H. He and X.-H. Wu, “Exp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700-708, 2006. · Zbl 1141.35448
[6] M. F. El-Sabbagh and A. T. Ali, “New exact solutions for (3+1)-dimensional Kadomtsev-Petviashvili equation and generalized (2+1)-dimensional Boussinesq equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 6, no. 2, pp. 151-162, 2005. · Zbl 1401.35267
[7] S. Liu, Z. Fu, S. Liu, and Q. Zhao, “Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69-74, 2001. · Zbl 0972.35062
[8] C. Dai and J. Zhang, “Jacobian elliptic function method for nonlinear differential-difference equations,” Chaos, Solitons & Fractals, vol. 27, no. 4, pp. 1042-1047, 2006. · Zbl 1091.34538
[9] M. F. El-Sabbagh, M. M. Hassan, and E. A.-B. Abdel-Salam, “Quasiperiodic waves and their interactions in the 2+1dimensional modified dispersive water-wave system,” Physica Scripta, vol. 80, no. 1, article 15006, 2009. · Zbl 1170.76310
[10] E. Fan, “Extended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212-218, 2000. · Zbl 1167.35331
[11] I. A. Hassanien, R. A. Zait, and E. A.-B. Abdel-Salam, “Multicnoidal and multitravelling wave solutions for some nonlinear equations of mathematical physics,” Physica Scripta, vol. 67, no. 6, pp. 457-463, 2003. · Zbl 1142.35569
[12] Y. Ren and H. Zhang, “New generalized hyperbolic functions and auto-Bäcklund transformation to find new exact solutions of the -dimensional NNV equation,” Physics Letters A, vol. 357, no. 6, pp. 438-448, 2006. · Zbl 1236.35004
[13] E. A. B. Abdel-Salam, “Periodic structures based on the symmetrical lucas function of the (2+1)-dimensional dispersive long-wave system,” Zeitschrift fur Naturforschung A, vol. 63, no. 10-11, pp. 671-678, 2008.
[14] E. A.-B. Abdel-Salam, “Quasi-periodic structures based on symmetrical Lucas function of (2+1)-dimensional modified dispersive water-wave system,” Communications in Theoretical Physics, vol. 52, no. 6, pp. 1004-1012, 2009. · Zbl 1382.35253
[15] E. A. B. Abdel-Salam and D. Kaya, “Application of new triangular functions to nonlinear partial differential equations,” Zeitschrift fur Naturforschung A, vol. 64, no. 1-2, pp. 1-7, 2009.
[16] E. A. B. Abdel-Salam, “Quasi-periodic, periodic waves, and soliton solutions for the combined KdV-mKdV equation,” Zeitschrift fur Naturforschung A, vol. 64, no. 9-10, pp. 639-645, 2009.
[17] E. A.-B. Abdel-Salam and Z. I. A. Al-Muhiameed, “Generalized Jacobi elliptic function method and non-travelling wave solutions,” Nonlinear Science Letters A, vol. 1, pp. 363-372, 2010.
[18] A. Borhanifar, M. M. Kabir, and L. M. Vahdat, “New periodic and soliton wave solutions for the generalized Zakharov system and (2+1)-dimensional Nizhnik-Novikov-Veselov system,” Chaos, Solitons and Fractals, vol. 42, no. 3, pp. 1646-1654, 2009. · Zbl 1198.35216
[19] J. L. Zhang and M. L. Wang, “Exact solutions to a class of nonlinear Schrödinger-type equations,” Pramana, vol. 67, no. 6, pp. 1011-1022, 2006.
[20] H. W. Schürmann, “Traveling-wave solutions of the cubic-quintic nonlinear Schrödinger equation,” Physical Review E, vol. 54, no. 4, pp. 4312-4320, 1996.
[21] Sirendaoreji and S. Jiong, “A direct method for solving sine-Gordon type equations,” Physics Letters A, vol. 298, no. 2-3, pp. 133-139, 2002. · Zbl 0995.35056
[22] Sirendaoreji and S. Jiong, “Auxiliary equation method for solving nonlinear partial differential equations,” Physics Letters A, vol. 309, no. 5-6, pp. 387-396, 2003. · Zbl 1011.35035
[23] Sirendaoreji, “A new auxiliary equation and exact travelling wave solutions of nonlinear equations,” Physics Letters A, vol. 356, no. 2, pp. 124-130, 2006. · Zbl 1160.35527
[24] M. Wang, X. Li, and J. Zhang, “Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation,” Physics Letters A, vol. 363, no. 1-2, pp. 96-101, 2007. · Zbl 1197.81129
[25] Z. I. A. Al-Muhiameed and E. A.-B. Abdel-Salam, “Generalized Jacobi elliptic function solution to a class of nonlinear Schrödinger-type equations,” Mathematical Problems in Engineering, vol. 2011, Article ID 575679, 11 pages, 2011. · Zbl 1217.35169
[26] E. A.-B. Abdel-Salam and Z. I. A. Al-Muhiameed, “Exotic localized structures based on the symmetrical lucas function of the (2+1)-dimensional generalized Nizhnik-Novikov-Veselov system,” Turkish Journal of Physics, vol. 35, pp. 1-16, 2011.
[27] J. L. Zhang, M. L. Wang, and X. Z. Li, “The subsidiary ordinary differential equations and the exact solutions of the higher order dispersive nonlinear Schrödinger equation,” Physics Letters A, vol. 357, no. 3, pp. 188-195, 2006. · Zbl 1236.81092
[28] Q. Wang and M. Tang, “New exact solutions for two nonlinear equations,” Physics Letters A, vol. 372, no. 17, pp. 2995-3000, 2008. · Zbl 1220.37069
[29] M. Wang, X. Li, and J. Zhang, “Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation,” Physics Letters A, vol. 363, no. 1-2, pp. 96-101, 2007. · Zbl 1197.81129
[30] E. J. Parkes, “Observations on the basic (G’/G) -expansion method for finding solutions to nonlinear evolution equations,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1759-1763, 2010. · Zbl 1203.35239
[31] W.-X. Ma and M. Chen, “Direct search for exact solutions to the nonlinear Schrödinger equation,” Applied Mathematics and Computation, vol. 215, no. 8, pp. 2835-2842, 2009. · Zbl 1180.65130
[32] M. Y. Moghaddam, A. Asgari, and H. Yazdani, “Exact travelling wave solutions for the generalized nonlinear Schrödinger (GNLS) equation with a source by extended tanh-coth, sine-cosine and exp-function methods,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 422-435, 2009. · Zbl 1173.35690
[33] R. Yang, R. Hao, L. Li, Z. Li, and G. Zhou, “Dark soliton solution for higher-order nonlinear Schrödinger equation with variable coefficients,” Optics Communications, vol. 242, no. 1-3, pp. 285-293, 2004.
[34] X. Li and M. Wang, “A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms,” Physics Letters A, vol. 361, no. 1-2, pp. 115-118, 2007. · Zbl 1170.35085
[35] S. Zhang, W. Wang, and J.-L. Tong, “The improved sub-ODE method for a generalized KdV-mKdV equation with nonlinear terms of any order,” Physics Letters A, vol. 372, no. 21, pp. 3808-3813, 2008. · Zbl 1220.35157
[36] M. M. Hassan and E. A.-B. Abdel-Salam, “New exact solutions of a class of higher-order nonlinear Schrödinger equations,” Journal of the Egyptian Mathematical Society, vol. 18, pp. 315-329, 2010. · Zbl 1362.34003
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.