Exact explicit solitary wave and periodic wave solutions and their dynamical behaviors for the Schamel-Korteweg-de Vries equation

doi:10.1088/1674-1056/abd7d1

Abstract The Schamel-Korteweg-de Vries equation is investigated by the approach of dynamics. The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses. The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given. All possible exact explicit parametric representations of the waves are also presented. Along with the details of the analyses, the analytical results are numerically simulated lastly.

Keywords: Schamel-Korteweg-de Vries equation dynamical behavior solitary wave solution periodic wave solution

Received: 19 November 2020
Revised: 28 December 2020
Accepted manuscript online: 04 January 2021

PACS:	02.30.Hq	(Ordinary differential equations)
	02.30.Oz	(Bifurcation theory)
	04.20.Jb	(Exact solutions)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11461022).

Corresponding Authors: Qing Meng
E-mail: mengqhhu@126.com

Cite this article:

Bin He(何斌) and Qing Meng(蒙清) Exact explicit solitary wave and periodic wave solutions and their dynamical behaviors for the Schamel-Korteweg-de Vries equation 2021 Chin. Phys. B 30 060201

[1] Gill T S, Bedi C and Saini N S 2011 Phys. Plasmas 18 043701
[2] Mushtaq A, Khattak M N, Ahmad Z and Qamar A 2012 Phys. Plasmas 19 042304
[3] Miura R M 1976 SIAM Review 18 412
[4] Dodd R K, Eilbeck J C, Gibbon J D and Morris H C 1982 Solitons and Nonlinear Wave Equations (London: Academic Press)
[5] Schamel H 1972 Plasma Phys. 14 905
[6] Schamel H and Shukla P K 1976 Phys. Rev. Lett. 36 968
[7] El-Kalaawy O H 2011 Phys. Plasmas 18 112302
[8] Ma W X and You Y C 2005 Trans. Am. Math. Soc. 357 1753
[9] Ma W X 2015 Phys. Lett. A 379 1975
[10] Ma W X and Zhou Y 2018 J. Differ. Equations 264 2633
[11] Khater A H, Hassan M M and Temsah R S 2005 J. Phys. Soc. Jpn. 74 1431
[12] Lee J and Sakthivel R 2011 Rep. Math. Phys. 68 153
[13] Hassan M M 2010 Commun. Theor. Phys. 53 596
[14] Wu Y and Liu Z R 2013 Abstr. Appl. Anal. 2013 483492
[15] Kangalgil F 2016 J. Egypt. Math. Soc. 24 526
[16] Giresunlu İ B, Saǧlam Özkan Y and Yasar E 2017 Math. Method Appl. Sci. 40 3927
[17] Daghan D and Donmez O 2018 J. Appl. Mech. Tech. Phys. 59 389
[18] Kengne E, Lakhssassi A and Liu W M 2020 Chaos Soliton Fract. 140 110229
[19] Li J B and Dai H H 2007 On the Study of Singular Nonlinear Traveling Wave Equation: Dynamical System Approach (Bejing: Science Press) (in Chinese)
[20] Li J B 2013 Singular Nonlinear Traveling Wave Equations: Bifurcations and Exact Solutions (Bejing: Science Press) (in Chinese)
[21] He B, Meng Q and Long Y 2012 Commun. Nonlinear Sci. Numer. Simulat. 17 4137
[22] He B and Meng Q 2016 Nonlinear Dyn. 86 811
[23] Li J B and Chen G R 2007 Int. J. Bifur. Chaos 17 4049
[24] Li J B, Zhang Y and Zhao X H 2009 Int. J. Bifur. Chaos 19 1995
[25] Byrd P F and Friedman M D 1971 Handbook of Elliptic Integrals for Engineers and Physicists (Berlin: Springer-Verlag)
[26] Chamdrasekharan K 1985 Elliptic Functions (Berlin: Springer-Verlag)

[1]	Dynamical analysis for hybrid virus infection system in switching environment Dong-Xi Li(李东喜), Ni Zhang(张妮). Chin. Phys. B, 2020, 29(9): 090201.
[2]	Dynamical behaviors of traveling wave solutions to a Fujimoto-Watanabe equation Zhen-Shu Wen(温振庶), Li-Juan Shi(师利娟). Chin. Phys. B, 2018, 27(9): 090201.
[3]	Exact transverse solitary and periodic wave solutions in a coupled nonlinear inductor-capacitor network Serge Bruno Yamgoué, Guy Roger Deffo, Eric Tala-Tebue, François Beceau Pelap. Chin. Phys. B, 2018, 27(9): 096301.
[4]	Quasi-periodic solutions and asymptotic properties for the nonlocal Boussinesq equation Zhen Wang(王振), Yupeng Qin(秦玉鹏), Li Zou(邹丽). Chin. Phys. B, 2017, 26(5): 050504.
[5]	A novel (G’/G)-expansion method and its application to the Boussinesq equation Md. Nur Alam, Md. Ali Akbar, Syed Tauseef Mohyud-Din. Chin. Phys. B, 2014, 23(2): 020203.
[6]	Nonautonomous solitary-wave solutions of the generalized nonautonomous cubic–quintic nonlinear Schrödinger equation with time- and space-modulated coefficients He Jun-Rong (何俊荣), Li Hua-Mei (李画眉). Chin. Phys. B, 2013, 22(4): 040310.
[7]	Folded localized excitations in the (2+1)-dimensional modified dispersive water-wave system Lei Yan (雷燕), Ma Song-Hua (马松华), Fang Jian-Ping (方建平). Chin. Phys. B, 2013, 22(1): 010506.
[8]	Riemann theta function periodic wave solutions for the variable-coefficient mKdV equation Zhang Yi (张翼), Cheng Zhi-Long (程智龙), Hao Xiao-Hong (郝晓红). Chin. Phys. B, 2012, 21(12): 120203.
[9]	Chaotic solutions of (2+1)-dimensional Broek–Kaup equation with variable coefficients Yang Zheng(杨征), Ma Song-Hua(马松华), and Fang Jian-Ping(方建平) . Chin. Phys. B, 2011, 20(4): 040301.
[10]	Combined periodic wave and solitary wave solutions in two-component Bose–Einstein condensates Yao Shu-Fang (姚淑芳), Li Qiu-Yan(李秋艳), and Li Zai-Dong(李再东) . Chin. Phys. B, 2011, 20(11): 110307.
[11]	Some exact solutions to the inhomogeneous higher-order nonlinear Schr?dinger equation by a direct method Zhang Huan-Ping(张焕萍), Li Biao(李彪), and Chen Yong(陈勇). Chin. Phys. B, 2010, 19(6): 060302.
[12]	A novel four-dimensional autonomous hyperchaotic system Liu Chong-Xin(刘崇新) and Liu Ling(刘凌). Chin. Phys. B, 2009, 18(6): 2188-2193.
[13]	New exact periodic solutions to (2+1)-dimensional dispersive long wave equations Zhang Wen-Liang(张文亮), Wu Guo-Jiang(吴国将), Zhang Miao(张苗), Wang Jun-Mao(王军帽), and Han Jia-Hua(韩家骅). Chin. Phys. B, 2008, 17(4): 1156-1164.
[14]	New periodic wave solutions, localized excitations and their interaction for 2+1-dimensional Burgers equation Ma Hong-Cai (马红彩), Ge Dong-Jie (葛东杰), Yu Yao-Dong (于耀东). Chin. Phys. B, 2008, 17(12): 4344-4353.
[15]	Discrete doubly periodic and solitary wave solutions for the semi-discrete coupled mKdV equations Wu Xiao-Fei(吴晓飞), Zhu Jia-Min(朱加民), and Ma Zheng-Yi(马正义). Chin. Phys. B, 2007, 16(8): 2159-2166.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Exact explicit solitary wave and periodic wave solutions and their dynamical behaviors for the Schamel-Korteweg-de Vries equation

Cite this article:

Online attention