The Modified Simple Equation Method, the Exp-Function Method, and the Method of Soliton Ansatz for Solving the Long–Short Wave Resonance Equations
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E.M.E. Zayed
Abstract
The modified simple equation method, the exp-function method, and the method of soliton ansatz for solving nonlinear partial differential equations are presented. Based on these three different methods, we obtain the exact solutions and the bright–dark soliton solutions with parameters of the long-short wave resonance equations which describe the resonance interaction between the long wave and the short wave. When these parameters take special values, the solitary wave solutions are derived from the exact solutions. We compare the results obtained using the three methods. Also, a comparison between our results and the well-known results is given.
Acknowledgments
The authors wish to thank the referees for their comments on this paper.
References
[1] A. M. Wazwaz, Appl. Math. Comput. 202, 275 (2008).10.1016/j.amc.2008.02.013Suche in Google Scholar
[2] J. H. He and X. H. Wu, Chaos Soliton Fract. 30, 700 (2006).10.1016/j.chaos.2006.03.020Suche in Google Scholar
[3] X. H. Wu and J. H. He, Comput. Math. Appl. 54, 966 (2007).10.1016/j.camwa.2006.12.041Suche in Google Scholar
[4] J.-H. He and L. N. Zhang, Phys. Lett. A 372, 1044 (2008).10.1016/j.physleta.2007.08.059Suche in Google Scholar
[5] S. D. Zhu, Int. J. Nonlinear Sci. Numer. Simul. 8, 465 (2007).Suche in Google Scholar
[6] S. Zhang, Chaos Soliton Fract. 38, 270 (2008).10.1016/j.chaos.2006.11.014Suche in Google Scholar
[7] D. D. Ganji, A. Asgari, and Z. Z. Ganji, Acta Appl. Math. 104, 201 (2008).10.1007/s10440-008-9252-0Suche in Google Scholar
[8] I. Aslan and V. Marinakis, Commun. Theor. Phys. 56, 397 (2011).10.1088/0253-6102/56/3/01Suche in Google Scholar
[9] I. Aslan, Commun. Theor. Phys. 60, 521 (2013).10.1088/0253-6102/60/5/01Suche in Google Scholar
[10] A. M. Wazwaz, Commun. Nonlinear Sci. Numer. Simul. 11, 148 (2006).10.1016/j.cnsns.2004.10.001Suche in Google Scholar
[11] A. M. Wazwaz, Phys. Lett. A 352, 500 (2006).10.1016/j.physleta.2005.12.036Suche in Google Scholar
[12] A. M. Wazwaz, Int. J. Comput. Math. 82, 235 (2005).10.1080/00207160412331296706Suche in Google Scholar
[13] E. Fan and H. Zhang, Phys. Lett. A 246, 403 (1998).10.1016/S0375-9601(98)00547-7Suche in Google Scholar
[14] E. M. E. Zayed and A. H. Arnous, Chin. Phys. Lett. 29, 080203 (2012).10.1088/0256-307X/29/8/080203Suche in Google Scholar
[15] W. Malfliet, Am. J. Phys. 60, 650 (1992).10.1119/1.17120Suche in Google Scholar
[16] W. Malfliet and W. Hereman, Phys. Scr. 54, 563 (1996).10.1088/0031-8949/54/6/003Suche in Google Scholar
[17] W. Malfliet and W. Hereman, Phys. Scr. 54, 569 (1996).10.1088/0031-8949/54/6/004Suche in Google Scholar
[18] A. M. Wazwaz, Chaos Soliton Fract. 25, 55 (2005).10.1016/j.chaos.2004.09.122Suche in Google Scholar
[19] A. M. Wazwaz, Appl. Math. Comput. 167, 210 (2005).10.1016/j.amc.2004.07.022Suche in Google Scholar
[20] E. Fan, Phys. Lett. A 277, 212 (2000).10.1016/S0375-9601(00)00725-8Suche in Google Scholar
[21] E. Fan and Y. C. Hon, Z. Naturforsch. 57a, 692 (2002).Suche in Google Scholar
[22] X. Zheng, Y. Chen, and H. Zhang, Phys. Lett. A 311, 145 (2003).10.1016/S0375-9601(03)00451-1Suche in Google Scholar
[23] A. M. Wazwaz, Commun. Nonlinear Sci. Numer. Simul. 13, 584 (2008).10.1016/j.cnsns.2006.06.014Suche in Google Scholar
[24] M. A. Abdou, Appl. Math. Comput. 190, 988 (2007).10.1016/j.amc.2007.01.070Suche in Google Scholar
[25] M. Wang, X. Li, and J. Zhang, Phys. Lett. A 372, 417 (2008).10.1016/j.physleta.2007.07.051Suche in Google Scholar
[26] E. M. E. Zayed and K. A. Gepreel, J. Math. Phys. 50, 013502 (2009).10.1063/1.3033750Suche in Google Scholar
[27] N. A. Kudryashov, Appl. Math. Comput. 217, 1755 (2010).10.1016/j.amc.2010.03.071Suche in Google Scholar
[28] I. Islan, Appl. Math. Comput. 217, 937 (2010).Suche in Google Scholar
[29] E. M. E. Zayed, J. Phys. A: Math. Theor. 42, 195202 (2009).10.1088/1751-8113/42/19/195202Suche in Google Scholar
[30] A. J. M. Jawad, M. D. Petkovic, and A. Biswas, Appl. Math. Comput. 217, 869 (2010).10.1016/j.amc.2010.06.030Suche in Google Scholar
[31] E. M. E. Zayed, Appl. Math. Comput. 218, 3962 (2011).10.1016/j.amc.2011.09.025Suche in Google Scholar
[32] E. M. E. Zayed and S. A. H. Ibrahim, Chin. Phys. Lett. 29, 060201 (2012).10.1088/0256-307X/29/6/060201Suche in Google Scholar
[33] E. M. E. Zayed and A. H. Arnous, Appl. Appl. Math. 8, 553 (2013).Suche in Google Scholar
[34] E. M. E. Zayed and A. H. Arnous, AIP Conf. Proc. 1479, 2044 (2012).Suche in Google Scholar
[35] W. X. Ma, T. Huang, and Y. Zhang, Phys. Script. 82, 065003 (2010).10.1088/0031-8949/82/06/065003Suche in Google Scholar
[36] E. M. E. Zayed and A.-G. Al-Nowehy, Z. Naturforsch. 70a, 775 (2015).10.1515/zna-2015-0151Suche in Google Scholar
[37] R. M. El-Shiekh and A.-G. Al-Nowehy, Z. Naturforsch. 68a, 255 (2013).10.5560/ZNA.2012-0108Suche in Google Scholar
[38] G. M. Moatimid, R. M. El-Shiekh, and A.-G. Al-Nowehy, Nonlinear Sci. Lett. A 4, 1 (2013).Suche in Google Scholar
[39] G. M. Moatimid, R. M. El-Shiekh, and A.-G. Al-Nowehy, Am. J. Comput. Appl. Math. 1, 1 (2011).Suche in Google Scholar
[40] E. M. E. Zayed, G. M. Moatimid, and A.-G. Al-Nowehy, Scientific J. Math. Res. 5, 19 (2015).Suche in Google Scholar
[41] G. M. Moatimid, R. M. El-Shiekh, and A.-G. Al-Nowehy, Appl. Math. Comput. 220, 455 (2013).10.1016/j.amc.2013.06.034Suche in Google Scholar
[42] M. H. M. Moussa and R. M. El-Sheikh, Physica A 371, 325 (2006).10.1016/j.physa.2006.04.044Suche in Google Scholar
[43] A. Biswas, D. Milovic, and M. Edwards, Mathematical Theory of Dispersion-Managed Optical Solitons, Springer-Verlag, New York 2010.10.1007/978-3-642-10220-2Suche in Google Scholar
[44] A. K. Sarma, M. Saha, and A. Biswas, J. Infrared Milli Terahz Waves 31, 1048 (2010).10.1007/s10762-010-9673-5Suche in Google Scholar
[45] Y. Shang, Chaos Soliton. Fract. 36, 762 (2008).10.1016/j.chaos.2006.07.007Suche in Google Scholar
[46] V. D. Djordjevic and L. G. Redekopp, J. Fluid Mech. 79, 703 (1977).10.1017/S0022112077000408Suche in Google Scholar
[47] A. Biswas, Commun. Nonlinear Sci. Numer. Simul. 15, 2744 (2010).10.1016/j.cnsns.2009.10.023Suche in Google Scholar
©2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Soliton, Breather, and Rogue Wave for a (2+1)-Dimensional Nonlinear Schrödinger Equation
- The Modified Simple Equation Method, the Exp-Function Method, and the Method of Soliton Ansatz for Solving the Long–Short Wave Resonance Equations
- Application of the Reverberation-Ray Matrix to the Non-Fourier Heat Conduction in Functionally Graded Materials
- Rational Solutions for Lattice Potential KdV Equation and Two Semi-discrete Lattice Potential KdV Equations
- Structural, Electronic, Magnetic and Optical Properties of Ni,Ti/Al-based Heusler Alloys: A First-Principles Approach
- Ab Initio Calculations on the Structural, Mechanical, Electronic, Dynamic, and Optical Properties of Semiconductor Half-Heusler Compound ZrPdSn
- Qualitative Behaviour of Generalised Beddington Model
- Studying Nuclear Level Densities of 238U in the Nuclear Reactions within the Macroscopic Nuclear Models
- Total π-Electron Energy of Conjugated Molecules with Non-bonding Molecular Orbitals
- Investigation of Thermal Expansion and Physical Properties of Carbon Nanotube Reinforced Nanocrystalline Aluminum Nanocomposite
- Bistable Bright Optical Spatial Solitons due to Charge Drift and Diffusion of Various Orders in Photovoltaic Photorefractive Media Under Closed-Circuit Conditions
- Application of a Differential Transform Method to the Transient Natural Convection Problem in a Vertical Tube with Variable Fluid Properties
Artikel in diesem Heft
- Frontmatter
- Soliton, Breather, and Rogue Wave for a (2+1)-Dimensional Nonlinear Schrödinger Equation
- The Modified Simple Equation Method, the Exp-Function Method, and the Method of Soliton Ansatz for Solving the Long–Short Wave Resonance Equations
- Application of the Reverberation-Ray Matrix to the Non-Fourier Heat Conduction in Functionally Graded Materials
- Rational Solutions for Lattice Potential KdV Equation and Two Semi-discrete Lattice Potential KdV Equations
- Structural, Electronic, Magnetic and Optical Properties of Ni,Ti/Al-based Heusler Alloys: A First-Principles Approach
- Ab Initio Calculations on the Structural, Mechanical, Electronic, Dynamic, and Optical Properties of Semiconductor Half-Heusler Compound ZrPdSn
- Qualitative Behaviour of Generalised Beddington Model
- Studying Nuclear Level Densities of 238U in the Nuclear Reactions within the Macroscopic Nuclear Models
- Total π-Electron Energy of Conjugated Molecules with Non-bonding Molecular Orbitals
- Investigation of Thermal Expansion and Physical Properties of Carbon Nanotube Reinforced Nanocrystalline Aluminum Nanocomposite
- Bistable Bright Optical Spatial Solitons due to Charge Drift and Diffusion of Various Orders in Photovoltaic Photorefractive Media Under Closed-Circuit Conditions
- Application of a Differential Transform Method to the Transient Natural Convection Problem in a Vertical Tube with Variable Fluid Properties
