New solitary wave solutions of the Korteweg-de Vries (KdV) equation by new version of the trial equation method

Yusuf Pandir; Ali Ekin

doi:10.61383/ejam.20231130

Authors

Yusuf Pandir Department of Mathematics, Faculty of Arts and Sciences, Yozgat Bozok University, 66100 Yozgat/Turkey Corresponding Author
Ali Ekin The Graduate School of Natural and Applied Sciences, Yozgat Bozok University, 66100 Yozgat, Turkey

DOI:

https://doi.org/10.61383/ejam.20231130

Keywords:

New version of the trial equation method, nonlinear partial differential equations, Korteweg-de Vries (KdV) equation, solitary wave soliton solutions

Abstract

New solitary wave solutions for the Korteweg-de Vries (KdV) equation by a new version of the trial equation method are attained. Proper transformation reduces the Korteweg-de Vries (KdV) equation to a quadratic ordinary differential equation that is fully integrated using the new version trial equation approach. The family of solitary wave solutions of the reduced equation ensures a combined expression for the Korteweg-de Vries (KdV) equation, which contains exact solutions derived in recent years using different integration methods. The analytic solution of the reduced equation permits to find exact solutions for the Korteweg-de Vries (KdV) equation, providing a variety of new solitary wave solutions that have not been reported before.

References

[1] W. Malfliet and W. Hereman, The Tanh method: I Exact solutions of nonlinear evolution and wave equations, Phys. Scripta, 54, 563-568, (1996). DOI: https://doi.org/10.1088/0031-8949/54/6/003

[2] H. A. Zedan, New approach for tanh and extended-tanh methods with applications on Hirota-Satsuma equations, Comput. Appl. Math. 28(1), 1-14, (2009). DOI: https://doi.org/10.1590/S1807-03022009000100001

[3] J. Hietarinta, Hirota’s bilinear method and its generalization, Int. J. Mod. Phys. A 12(1), 43-51 (1997). DOI: https://doi.org/10.1142/S0217751X97000062

[4] C.A. G´ omez, A. Jhangeer, H. Rezazadeh, R.A. Talarposhti and A. Bekir, Closed form solutions of the perturbed Gerdjikov-Ivanov equation with variable coefficients, East Asian J. Appl. M 11(1), 207-218 (2021). DOI: https://doi.org/10.4208/eajam.230620.070920

[5] W. X. Ma, Y. Zhang and Y. Tang, Symbolic computation of lump solutions to a combined equation involving three types of nonlinear terms, East Asian J. Appl. M 10(4),732-745(2020) DOI: https://doi.org/10.4208/eajam.151019.110420

[6] J. H. He and X. H. Wu, Exp-function method for nonlinear wave equations, Chaos, Soliton Fract. 30, 700-708 (2006). DOI: https://doi.org/10.1016/j.chaos.2006.03.020

[7] L.K. Ravi, S. S. Ray and S. Sahoo, New exact solutions of coupled BoussinesqBurgers equations by Exp-function method, J. Ocean Eng. Sci 2, 34-46 (2017). DOI: https://doi.org/10.1016/j.joes.2016.09.001

[8] M. A. Akbar, N. H. M. Ali and S. T. Mohyud-Din, The modified alternative (G′/G)- expansion method to nonlinear evolution equation: application to the (1+1)dimensional Drinfel’d-Sokolov-Wilson equation, SpringerPlus 327, 2-16 (2013). DOI: https://doi.org/10.1186/2193-1801-2-327

[9] M. Shakeel and S. T. Mohyud-Din, New (G′/G)-expansion method and its application to the Zakharov Kuznetsov-Benjamin-Bona-Mahony ( KK- BBM) equation, J. Assoc. Arab Univ. Basic Appl. Sci. 18(1), 66-81 (2015). DOI: https://doi.org/10.1016/j.jaubas.2014.02.007

[10] C. S. Liu, Trial equation method for nonlinear evolution equations with rank inhomogeneous: mathematical discussions and applications, Commun. Theor. Phys. 45(2), 219-223 (2006). DOI: https://doi.org/10.1088/0253-6102/45/2/005

[11] C. S. Liu, Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations, Comput. Phys. Commun. 181(2), 317-324 (2010). DOI: https://doi.org/10.1016/j.cpc.2009.10.006

[12] Y. Gurefe, A. Sonmezoglu and E. Misirli, Application of trial equation method to the nonlinear partial differen tial equations arising in mathematical physics, Pramana-J. Phys. 77(6), 1023-1029 (2011). DOI: https://doi.org/10.1007/s12043-011-0201-5

[13] Y. Gurefe, A. Sonmezoglu and E. Misirli, Application of an irrational trial equation method to high dimensional nonlinear evolution equations, J. Adv. Math. Stud. 5(1), 41-47 (2012).

[14] Y. Pandir, Y. Gurefe, U. Kadak and E. Misirli, Classifications of exact solutions for some nonlinear partial differential equations with generalized evolution, Abstr. Appl. Anal. 2012, 1-16 (2012). DOI: https://doi.org/10.1155/2012/478531

[15] Y. Pandir, Y. Gurefe and E. Misirli, Classification of exact solutions to the generalized Kadomtsev-Petviashvili equation, Phys. Scr. 87(2), 1-12 (2013). DOI: https://doi.org/10.1088/0031-8949/87/02/025003

[16] Y. Gurefe, E. Misirli, A. Sonmezoglu and M. Ekici, Extended trial equation method to generalized nonlinear partial differential equations, Appl. Math. Comput. 219(10), 5253-5260 (2013). DOI: https://doi.org/10.1016/j.amc.2012.11.046

[17] J. Zhang, F. Jiang and X. Zhao, An improved (G′/G)-expansion method for solving nonlinear evolution equations, Int. J. Comput. Math. 87(8), 1716-1725 (2010). DOI: https://doi.org/10.1080/00207160802450166

[18] S. Guo and Y. Zhou, The extended (G′/G)-expansion method and its applications to the Whitham-Broer Kaup-Like equations and coupled Hirota-Satsuma KdV equations, Appl. Math. Comput. 215, 3214-3221 (2010). DOI: https://doi.org/10.1016/j.amc.2009.10.008

[19] Y. Pandir, Y. Gurefe and E. Misirli, A multiple extended trial equation method for the fractional Sharma-Tasso Olver equation, AIP Conf. Proc. 1558, 1927 (2013). DOI: https://doi.org/10.1063/1.4825910

[20] X. J. Laia, J. F. Zhang and S. H. Meia, Application of the Weierstrass elliptic expansion method to the long-wave and short-wave resonance interaction system, Z. Naturforsch. 63a, 273-279 (2008). DOI: https://doi.org/10.1515/zna-2008-5-606

[21] Z.Y.Zhang,Jacobielliptic function expansion methodforthemodifiedKorteweg-deVries-Zakharov-Kuznetsov and the Hirota equations, Rom. J. Phys. 60(9-10), 13841394(2015)

[22] S. Shikuo, F. Zuntao, L. Shida and Z. Qiang, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A 289, 69-74 (2001). DOI: https://doi.org/10.1016/S0375-9601(01)00580-1

[23] A. Bekir and O. Unsal, Analytic treatment of nonlinear evolution equations using first integral method, Pramana-J. Phys. 79(1), 3-17 (2012). DOI: https://doi.org/10.1007/s12043-012-0282-9

[24] Y. Pandir, Symmetric Fibonacci function solutions of some nonlinear partial differential equations, Appl. Math. Inf. Sci. 8, 2237-2241 (2014). DOI: https://doi.org/10.12785/amis/080518

[25] Y. A. Tandogan, Y. Pandir and Y. Gurefe, Solutions of the nonlinear differential equations by use of modified Kudryashov method, Turkish J. Math. Comput. Sci. 1, 54-60 (2013).

[26] Y. Pandir and N. Turhan, A new version of the generalized F-expansion method and its applications, AIP Conf. Proc. 1798, 020122 (2017). DOI: https://doi.org/10.1063/1.4972714

[27] Y. Pandir, A new type of the generalized F-expansion method and its application to Sine-Gordon equation, Celal Bayar Univ. J. Sci. 13(3),647-650 (2017). DOI: https://doi.org/10.18466/cbayarfbe.306899

[28] Y. Pandir, S. T. Demiray and H. Bulut, A new approach for some NLDEs with variable coefficients, Optik 127, 11183-11190 (2016). DOI: https://doi.org/10.1016/j.ijleo.2016.08.019

[29] S. T. Demiray, Y. Pandir and H. Bulut, New solitary wave solutions of Maccari system, Ocean Eng. 103, 153-159 (2015). DOI: https://doi.org/10.1016/j.oceaneng.2015.04.037

[30] S. T. Demiray, Y. Pandir and H. Bulut, New soliton solutions for Sasa-Satsuma equation, Waves Random Complex Media 25(3), 417-418 (2015). DOI: https://doi.org/10.1080/17455030.2015.1042945

[31] Y. Pandir, A. Sonmezoglu, H. H. Duzgun and N. Turhan, Exact solutions of nonlinear Schr¨ odinger’s equation by using generalized Kudryashov method, AIP Conf. Proc. 1648,370004(2015). DOI: https://doi.org/10.1063/1.4912593

[32] Y. Pandir and A. Ekin, Dynamics of combined soliton solutions of unstable nonlinear Schrodinger equation with new version of the trial equation method, Chinese J. Physics 67, 534-543 (2020). DOI: https://doi.org/10.1016/j.cjph.2020.08.013

[33] K. Yang and J. Liu, The extended F-expansion method and exact solutions of nonlinear PDEs, Chaos, Solitons Fract. 22, 111-121 (2014). DOI: https://doi.org/10.1016/j.chaos.2003.12.069

[34] M. A. Abdou, Further improved F-expansion and new exact solutions for nonlinear evolution equations, Nonlinear Dyn. 52, 277-288 (2008). DOI: https://doi.org/10.1007/s11071-007-9277-3

[35] M. M. El-Borai, H. M. El-Owaidy, H. M. Ahmed, A. H. Arnous, S. Moshokoa, A. Biswasand M Belic, Topological and singular soliton solution to Kundu-Eckhaus equation with extended Kudryashov’s method, Optik, 128, 57-62 (2017). DOI: https://doi.org/10.1016/j.ijleo.2016.10.011

[36] W. X. MaandB. Fuchssteiner, Explicit and exact solutions to a Kolmogrov-PetrovskiPiskunov equation, Int. J. Nonlinear Mech. 31(3), 329-338 (1996). DOI: https://doi.org/10.1016/0020-7462(95)00064-X

[37] C. S. Liu, Trial equation method and its applications to nonlinear evolution equations, Phys. Sinica 54(6), 2505-2509 (2005). DOI: https://doi.org/10.7498/aps.54.2505

[38] C. S. Liu, Trial equation to solve the exact solutions for two kinds of KdV equations with variable coeffients, Acta Phys. Sinica 54, 4506-4510 (2005). DOI: https://doi.org/10.7498/aps.54.4506