New exact solutions for the Kaup-Kupershmidt equation

Mustafa Inc; Mamun Miah; Akher Chowdhury; Shahadat Ali; Hadi Rezazadeh; Mehmet Ali Akinlar; Yu-Ming Chu; Mustafa Inc; Mamun Miah; Akher Chowdhury; Shahadat Ali; Hadi Rezazadeh; Mehmet Ali Akinlar; Yu-Ming Chu

doi:10.3934/math.2020432

AIMS Mathematics

2020, Volume 5, Issue 6: 6726-6738. doi: 10.3934/math.2020432

Previous Article Next Article

Research article

New exact solutions for the Kaup-Kupershmidt equation

1.
Department of Mathematics, Firat University, Elazig, Turkey
2.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
3.
Department of Mathematics, Khulna University of Engineering and Technology, Bangladesh
4.
Department of Mathematics, Bangladesh Army University of Engineering and Technology, Bangladesh
5.
Department of Applied Mathematics, Noakhali Science and Technology University, Bangladesh
6.
Faculty of Engineering Technology, Amol University of Special Modern Technology, Amol, Iran
7.
Department of Mathematical Engineering, Yildiz Technical University, Istanbul, Turkey
8.
Department of Mathematics, Huzhou University, Huzhou 313000, China
9.
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science & Technology, Changsha 410114, China

Received: 23 June 2020 Accepted: 23 August 2020 Published: 28 August 2020
MSC : 35A09, 35E05

We present new exact solutions for the (1+1)-dimensional Kaup-Kupershmidt (KK) equation by employing method of double $(G'/G, 1/G)$-expansion. We express solutions by hyperbolic, trigonometric and rational functions explicitly. Computational results indicate the efficiency and applicability potential of the method.
- Kaup-Kupershmidt equation,
- the method of double $(G'/G,1/G)$-expansion,
- exact solutions
Citation: Mustafa Inc, Mamun Miah, Akher Chowdhury, Shahadat Ali, Hadi Rezazadeh, Mehmet Ali Akinlar, Yu-Ming Chu. New exact solutions for the Kaup-Kupershmidt equation[J]. AIMS Mathematics, 2020, 5(6): 6726-6738. doi: 10.3934/math.2020432

Related Papers:

Abstract

We present new exact solutions for the (1+1)-dimensional Kaup-Kupershmidt (KK) equation by employing method of double $(G'/G, 1/G)$-expansion. We express solutions by hyperbolic, trigonometric and rational functions explicitly. Computational results indicate the efficiency and applicability potential of the method.

References

[1]	C. Yue, D. Lu, M. M. A. Khater, et al. On explicit wave solutions of the fractional nonlinear DSW system via the modified Khater method, Fractals, 2020.
[2]	C. Yue, M. M. A. Khater, M. Inc, et al. Abundant analytical solutions of the fractional nonlinear (2+1)-dimensional BLMP equation arising in incompressible fluid, Int. J. Mod. Phys. B, 34 (2020), 1-13.
[3]	N. Mahak, G. Akram, Exact solitary wave solutions of the (1+1)-dimensional Fokas-Lenells equation, Optik, 208 (2020), 1-9.
[4]	H. Rezazadeh, J. Manafian, F. S. Khodadad, et al. Traveling wave solutions for density-dependent conformable fractional diffusion-reaction equation by the first integral method and the improvedtan(1/2φ(ξ))-expansion method, Opt. Quant. Electron., 50 (2018), 1-15. doi: 10.1007/s11082-017-1266-2
[5]	H. Rezazadeh, J. Vahidi, A. Zafar, et al. The functional variable method to find new exact solutions of the nonlinear evolution equations with dual-power-law nonlinearity, Int. J. Nonlin. Sci. Num., 21 (2019), 249-257.
[6]	B. Soltanalizadeh, H. Esmalifalak, R. Hekmati, et al. Numerical analysis of the one-demential wave equation subject to a boundary integral specification, WJST., 15 (2018), 421-437.
[7]	Z. Sarmast, B. Soltanalizadeh, K. Boubaker, A new numerical method to study a Second-order hyperbolic equation, South Asian Journal of Mathematics, 4 (2014), 285-296.
[8]	M. D. Hossain, M. K. Alam, M. A. Akbar, Abundant wave solutions of the Boussinesq equation and the (2+1)-dimensional extended shallow water wave equation, Ocean Engineering, 165 (2018), 69-76. doi: 10.1016/j.oceaneng.2018.07.025
[9]	M. D. Hossain, U. Kulsum, M. K. Alam, et al. Kink and periodic solutions to the Jimbo-Miwa equation and the Calogero-Bogoyavlenskii-Schiff equation, J. Mech. Cont. Math. Sci., 13 (2018), 50-66.
[10]	S. T. A. Siddique, M. D. Hossain, M. A. Akbar, Exact wave solutions to the (2+1)-dimensional Klein-Gordon equation with special types of nonlinearity, J. Mech. Cont. Math. Sci., 14 (2019), 1-20.
[11]	N. Sajid, G. Akram, Novel solutions of Biswas-Arshed equation by newly φ6-model expansion method, Optik, 211 (2020), 1-22.
[12]	Y. Chen, Q. Wang, Extended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic functions to (1+1)-dimensional dispersive long wave equation, Chaos Soliton. Fract., 24 (2005), 745-757. doi: 10.1016/j.chaos.2004.09.014
[13]	D. Lü, Jacobi elliptic function solutions for two variant Boussinesq equations, Chaos Soliton. Fract., 24 (2005), 1373-1385. doi: 10.1016/j.chaos.2004.09.085
[14]	A. Korkmaz, O. E. Hepson, K. Hosseini, et al. Sine-Gordon expansion method for exact solutions to conformable time fractional equations in RLW-class, J. King Saud Univ. Sci., 32 (2018), 567-574.
[15]	A. Chen, J. Li, Single peak solitary wave solutions for the osmosis K(2,2) equation under inhomogeneous boundary condition, J. Math. Anal. Appl., 369 (2010), 758-766. doi: 10.1016/j.jmaa.2010.04.018
[16]	D. H. Feng, J. B. Li, Exact explicit Traveling wave solutions for the (n+1)-dimensional Ø⁶ field model, Phys. Lett. A, 369 (2007), 255-261. doi: 10.1016/j.physleta.2007.04.088
[17]	J. H. He, X. H. Wu, Exp-function method for nonlinear wave equations, Chaos Soliton. Fract., 30 (2006), 700-708. doi: 10.1016/j.chaos.2006.03.020
[18]	A. Bekir, Application of the exp-function method for nonlinear differential-difference equations, Appl. Math. Comput., 215 (2010), 4049-4053.
[19]	H. Rezazadeh, A. Korkmaz, M. Eslami, et al. A large family of optical solutions to Kundu-Eckhaus model by a new auxiliary equation method, Opt. Quant. Electron., 51 (2019), 1-12. doi: 10.1007/s11082-018-1712-9
[20]	N. Raza, M. R. Aslam, H. Rezazadeh, Analytical study of resonant optical solitons with variable coefficients in Kerr and non-Kerr law media, Opt. Quant. Electron., 51 (2019), 59.
[21]	N. Raza, U. Afzal, A. R. Butt, et al. Optical solitons in nematic liquid crystals with Kerr and parabolic law nonlinearities, Opt. Quant. Electron., 51 (2019), 1-16. doi: 10.1007/s11082-018-1712-9
[22]	D. Feng, K. Li, On exact traveling wave solutions for (1+1)-dimensional Kaup-Kupershmidt equation, Appl. Math., 2 (2011), 752-756. doi: 10.4236/am.2011.26100
[23]	F. Batool, G. Akram, Application of extended Fan sub-equation method to (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mohony equation with fractional evolution, Opt. Quant. Electron., 49 (2017), 1-9. doi: 10.1007/s11082-016-0848-8
[24]	M. A. Fiddy, M. Testorf, Inverse scattering method applied to the synthesis of strongly structures, Opt. Express, 14 (2006), 2037-2046. doi: 10.1364/OE.14.002037
[25]	H. M. S. Ali, M. A. Habib, M. M. Miah, et al. A modification of the generalized Kudryshov method for the system of some nonlinear evolution equations, J. Mech. Cont. Math. Sci., 14 (2019), 91-109.
[26]	M. M. Rahman, M. A. Habib, H. M. S. Ali, et al. The generalized Kudryshov method: a renewed mechanism for performing exact solitary wave solutions of some NLEEs, J. Mech. Cont. Math. Sci., 14 (2019), 323-339.
[27]	M. A. Habib, H. M. S. Ali, M. M. Miah, et al. The generalized Kudryashov method for new closed form traveling wave solutions to some NLEEs, AIMS Mathematics, 4 (2019), 896-909. doi: 10.3934/math.2019.3.896
[28]	A. M. Wazwaz, The Hirota's bilinear method and the tanh-coth method for multiple-soliton solutions of the Sawada-Kotera-Kadomstsev-Petviashvili equation, Appl. Math. Computat., 200 (2008), 160-166. doi: 10.1016/j.amc.2007.11.001
[29]	J. G. Liu, M. Eslami, H. Rezazadeh, et al. Rational solutions and lump solutions to a non-isospectral and generalized variable-coefficient Kadomtsev-Petviashvili equation, Nonlinear Dynam., 95 (2019), 1027-1033. doi: 10.1007/s11071-018-4612-4
[30]	W. Gao, H. Rezazadeh, Z. Pinar, et al. Novel explicit solutions for the nonlinear Zoomeron equation by using newly extended direct algebraic technique, Opt. Quant. Electron., 52 (2020), 1-13. doi: 10.1007/s11082-019-2116-1
[31]	H. Jafari, N. Kadkhoda, D. Baleanu, Fractional Lie group method of the time-fractional Boussinesq equation, Nonlinear Dynam., 81 (2015), 1569-1574. doi: 10.1007/s11071-015-2091-4
[32]	L. H. Ling, L. X. Qiang, Exact Solutions to (2+1)-dimensional Kaup-Kupershmidt equation, Commun. Theor. Phys., 52 (2009), 795-800. doi: 10.1088/0253-6102/52/5/06
[33]	A. H. Bhrawy, A. Bishwas, M. Javidi, et al. New solutions for (1+1)-dimensional and (2+1)- dimensional Kaup-Kupershmidt equations, Results Math., 63 (2013), 675-686. doi: 10.1007/s00025-011-0225-7
[34]	M. A. Akbar, N. H. M. Ali, E. M. E. Zayed, Abundant exact traveling wave solutions of the generalized Bretherton equation via (G'/G)-expansion method, Commun. Theor. Phys., 57 (2012), 173-178. doi: 10.1088/0253-6102/57/2/01
[35]	B. Ayhan, A. Bekir, The (G'/G)-expansion method for the nonlinear lattice equations, Commun. Nonlinear Sci., 17 (2012), 3490-3498. doi: 10.1016/j.cnsns.2012.01.009
[36]	M. Wang, X. Li, J. Zhang, The (G'/G)-expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics, Phys. Lett. A, 372 (2008), 417-423. doi: 10.1016/j.physleta.2007.07.051
[37]	N. A. Kudryashov, A note on the (G'/G)-expansion method, Appl. Math. comput., 217 (2010), 1755-1758.
[38]	L. Li, E. Li, M. Wang, The (G'/G, 1/G)-expansion method and its application to traveling wave solutions of the Zakharov equations, Appl. Math. J. Chin. Univ., 25 (2010), 454-462.
[39]	E. M. E. Zayed, K. A. E. Alurrfi, The (G'/G, 1/G)-expansion method and its applications for solving two higher order nonlinear evolution equations, Math. Probl. Eng., 2014 (2014), 1-20.
[40]	M. M. Miah, H. M. S. Ali, M. A. Akbar, An investigation of abundant traveling wave solutions of complex nonlinear evolution equations: the perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg-Landau equation, Cogent Mathematics, 3 (2016), 1-19.
[41]	M. M. Miah, H. M. S. Ali, M. A. Akbar, et al. Some applications of the (G'/G, 1/G)-expansion method to find new exact solutions of NLEEs, Eur. Phys. J. Plus, 132 (2017), 1-15.
[42]	H. M. S. Ali, M. M. Miah, M. A. Akbar, Study of abundant explicit wave solutions of the Drinfeld-Sokolov-Satsuma-Hirota (DSSH) equation and the shallow water wave equation, Propulsion and Power Research, 7 (2018), 320-328. doi: 10.1016/j.jppr.2018.11.007
[43]	E. M. E. Zayed, K. A. E. Alurrfi, The (G'/G, 1/G)-expansion method and its applications to two nonlinear Schrödinger equations describing the propagation of femtosecond pulses in nonlinear optical fibers, Optik, 127 (2016), 1581-1589.
[44]	M. M. Miah, H. M. S. Ali, M. A. Akbar, et al. New applications of the two variable (G'/G, 1/G)- expansion method for closed form traveling wave solutions of Integro-differential equations, Journal of Ocean Engineering and Science, 4 (2019), 132-143.
[45]	M. M. Miah, A. R. Seadawy, H. M. S. Ali, et al. Further investigations to extract abundant new exact traveling wave solutions of some NLEEs, Journal of Ocean Engineering and Science, 4 (2019), 387-394. doi: 10.1016/j.joes.2019.06.004

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)