×
Zayed, Elsayed M. E.; Al-Nowehy, Abdul-Ghani

Exact solutions and optical soliton solutions for the nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity. (English) Zbl 1386.78017

Ric. Mat. 66, No. 2, 531-552 (2017).
Summary: In this article, we apply the soliton ansatz method combined with the Jacobi elliptic equation method which is different from the F-expansion method to obtain several types of Jacobi elliptic function solutions, the optical bright-dark-singular soliton solutions and trigonometric function solutions of the nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity, self-steeping and self-frequency shift effects which describes the propagation of an optical pulse in optical fibers. Comparison between our results in this article and the well-known results are given.

Cited in 2 Documents

MSC:

78A60 Lasers, masers, optical bistability, nonlinear optics
35Q51 Soliton equations
35Q55 NLS equations (nonlinear Schrödinger equations)

Keywords:

soliton ansatz method combined with the Jacobi elliptic equation method; F-expansion method; exact solutions; Jacobi elliptic function solutions; optical soliton solutions; nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity
Cite Review PDF
Full Text: DOI

References:

[1] Wazwaz, A.M.: Solitary wave solutions of the generalized shallow water wave (GSWW) equation by Hirota’s method, tanh-coth method and exp-function method. Appl. Math. Comput. 202, 275-286 (2008) · Zbl 1147.65109
[2] He, J.H., Wu, X.H.: Exp-function method for nonlinear wave equations. Chaos Solitons Fract. 30, 700-708 (2006) · Zbl 1141.35448 · doi:10.1016/j.chaos.2006.03.020
[3] Zayed, E.M.E., Al-Nowehy, Abdul-Ghani: Exact solutions for nonlinear foam drainage equation. Indian J. Phys. 91, 209-218 (2017) · doi:10.1007/s12648-016-0911-0
[4] Wazwaz, A.M.: A sine-cosine method for handling nonlinear wave equations. Math Comput. Model. 40, 499-508 (2004) · Zbl 1112.35352 · doi:10.1016/j.mcm.2003.12.010
[5] Yusufoğlu, E., Bekir, A., Alp, M.: Periodic and solitary wave solutions of Kawahara and modified Kawahara equations by using sine-cosine method. Chaos Solitons Fract. 37, 1193-1197 (2008) · Zbl 1148.35351 · doi:10.1016/j.chaos.2006.10.012
[6] Zayed, E.M.E., Al-Nowehy, Abdul-Ghani: Solitons and other solutions for the generalized KdV-mKdV equation with higher-order nonlinear terms. J. Partial Differ. Equ. 29, 218-245 (2016) · Zbl 1374.35107 · doi:10.4208/jpde.v29.n3.5
[7] Fan, E., Zhang, H.: A note on the homogeneous balance method. Phys. Lett. A 246, 403-406 (1998) · Zbl 1125.35308 · doi:10.1016/S0375-9601(98)00547-7
[8] Zayed, E.M.E., Arnous, A.H.: DNA dynamics studied using the homogeneous balance method. Chin. Phys. Lett. 29, 080203-080205 (2012) · doi:10.1088/0256-307X/29/8/080203
[9] Malfliet, W., Hereman, W.: The tanh method: II. Perturbation technique for conservative systems. Phys. Scr. 54, 569-575 (1996) · Zbl 0942.35035 · doi:10.1088/0031-8949/54/6/004
[10] Wazwaz, A.M.: The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations. Appl. Math. Comput. 167, 210-1196 (2005) · Zbl 1082.65585
[11] Fan, E., Hon, Y.C.: Generalized tanh method extended to special types of nonlinear equations. Z. Naturforsch. 57a, 692-700 (2002)
[12] Abdou, M.A.: The extended tanh method and its applications for solving nonlinear physical models. Appl. Math. Comput. 190, 988-996 (2007) · Zbl 1123.65103
[13] Wang, M., Li, X., Zhang, J.: The \[(G^{\prime }/G)\](G′/G)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics. Phys. Lett. A 372, 417-423 (2008) · Zbl 1217.76023 · doi:10.1016/j.physleta.2007.07.051
[14] Zayed, E.M.E., Gepreel, K.A.: The \[(G^{\prime }/G)\](G′/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations in mathematical physics. J. Math. Phys. 50, 013502-013513 (2009) · doi:10.1063/1.3033750
[15] Kudryashov, N.A.: A note on the \[(G^{\prime }/G)\](G′/G)-expansion method. Appl. Math. Comput. 217, 1755-1758 (2010) · Zbl 1203.35228
[16] Zayed, E.M.E.: New traveling wave solutions for higher dimensional nonlinear evolution equations using a generalized \[(G^{\prime }/G)\](G′/G)-expansion method. J. Phys. A Math. Theor. 42, 195202-195214 (2009) · Zbl 1170.35310 · doi:10.1088/1751-8113/42/19/195202
[17] Jawad, A.J.M., Petkovic, M.D., Biswas, A.: Modified simple equation method for nonlinear evolution equations. Appl. Math. Comput. 217, 869-877 (2010) · Zbl 1201.65119
[18] Zayed, E.M.E.: A note on the modified simple equation method applied to Sharma-Tasso-Olver equation. Appl. Math. Comput. 218, 3962-3964 (2011) · Zbl 1239.35170
[19] Zayed, E.M.E., Al-Nowehy, Abdul-Ghani: The modified simple equation method, the exp-function method and the method of soliton ansatz for solving the long-short wave resonance equations. Z. Naturforsch. 71a, 103-112 (2016)
[20] Ma, W.X., Huang, T., Zhang, Y.: A multiple exp-function method for nonlinear differential equations and its application. Phys. Scr. 82, 065003 (2010) · Zbl 1219.35209 · doi:10.1088/0031-8949/82/06/065003
[21] Zayed, E.M.E., Al-Nowehy, Abdul-Ghani: The multiple exp-function method and the linear superposition principle for solving the (2+1)-Dimensional Calogero-Bogoyavlenskii-Schiff equation. Z. Naturforsch. 70a, 775-779 (2015)
[22] El-Shiekh, R.M., Al-Nowehy, Abdul-Ghani: Integral methods to solve the variable coefficient NLSE. Z. Naturforsch. 68a, 255-260 (2013) · doi:10.5560/ZNA.2012-0108
[23] Moatimid, G.M., El-Shiekh, R.M., Al-Nowehy, Abdul-Ghani: New exact solutions for the variable coefficient two-dimensional Burger equation without restrictions on the variable coefficient. Nonlinear Sci. Lett. A 4, 1-7 (2013)
[24] Kudryashov, N.A.: One method for finding exact solutions of nonlinear differential equations. Commun. Nonlinear Sci. Numer. Simul. 17, 2248-2253 (2012) · Zbl 1250.35055 · doi:10.1016/j.cnsns.2011.10.016
[25] Zayed, E.M.E., Moatimid, G.M., Al-Nowehy, Abdul-Ghani: The generalized Kudryashov method and its applications for solving nonlinear PDEs in mathematical physics. Sci. J. Math. Res. 5, 19-39 (2015)
[26] Moatimid, G.M., El-Shiekh, R.M., Al-Nowehy, Abdul-Ghani A.A.H.: Exact solutions for Calogero-Bogoyavlenskii-Schiff equation using symmetry method. Appl. Math. Comput. 220, 455-462 (2013) · Zbl 1329.35271
[27] Moussa, M.H.M., El-Shiekh, R.M.: Similarity reduction and similarity solutions of Zabolotskay-Khoklov equation with dissipative term via symmetry method. Phys. A 371, 325-335 (2006) · doi:10.1016/j.physa.2006.04.044
[28] Sarma, A.K., Saha, M., Biswas, A.: Optical solitons with power law nonlinearity and hamiltonian perturbations: an exact solution. J. Infrared Milli Terahz Waves 31, 1048-1056 (2010) · doi:10.1007/s10762-010-9673-5
[29] Biswas, A.: 1-soliton solution of Benjamin-Bona-Mahoney equation with dual-power law nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 15, 2744-2746 (2010) · Zbl 1222.35157 · doi:10.1016/j.cnsns.2009.10.023
[30] Cevikel, A.C., Aksoy, E., Güner, Ö., Bekir, A.: Dark-bright soliton solutions for some evolution equations. Int. J. Nonlinear Sci. 16, 195-202 (2013) · Zbl 1394.35093
[31] Zhou, Q., Zhu, Q., Savescu, M., Bhrawy, A., Biswas, A.: Optical solitons with nonlinear dispersion in parabolic law medium. Proc. Rom. Acad. Ser. A 16, 152-159 (2015)
[32] Zayed, E.M.E., Al-Nowehy, Abdul-Ghani: Exact solutions and optical soliton solutions for the \[(2+12+1)\]-dimensional hyperbolic nonlinear Schr ödinger equation. Optik 127, 4970-4983 (2016) · doi:10.1016/j.ijleo.2016.02.010
[33] Zhang, J., Dai, C.: Bright and dark optical solitons in the nonlinear Schrödinger equation with fourth-order dispersion and cubic-quintic nonlinearity. Chin. Opt. Lett. 3, 295-298 (2005)
[34] Hassan, M.M., Abdel-Wahab, N.H., Abdel-Daym, M.S.: Exact solutions of the higher-order nonlinear Schrodinger equation with cubic-quintic nonlinearities, self-steeping and self-frequency shift effects. Int. J. Pure Appl. Math. 106, 495-513 (2016) · doi:10.12732/ijpam.v106i2.13
[35] Zayed, E.M.E., Amer, Y.A., Shohib, R.M.A.: The Jacobi elliptic function expansion method and its applications for solving the higher order dispersive nonlinear Schrödinger equation. Sci. J. Math. Res. 4, 53-72 (2014)
[36] Zhou, Q., Yao, D.Z., Chen, F.: Analytical study of optical solitons in media with Kerr and parabolic-law nonlinearities. J. Mod. Opt. 60, 1652-1657 (2013) · doi:10.1080/09500340.2013.852695
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.