Bifurcations and exact traveling wave solutions of the (n+1)-dimensional q-deformed double sinh-Gordon equations

Jie Song

doi:10.3934/dcdss.2022113

Article Contents

2023, Volume 16, Issue 3&4: 573-588. Doi: 10.3934/dcdss.2022113

This issue Previous Article Non-local dispersal equations with almost periodic dependence. II. Asymptotic dynamics of Fisher-KPP equations Next Article Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential

Bifurcations and exact traveling wave solutions of the (n+1)-dimensional q-deformed double sinh-Gordon equations

Jie Song^1,2,

1.
School of Economics and Finance, Huaqiao University, Quanzhou, Fujian 362021, China
2.
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China

Received: March 2022

Revised: April 2022

Early access: May 2022

Published: March 2023

Abstract / Introduction Full Text(HTML) Figure(15) Related Papers Cited by

Abstract

We consider an (n+1)-dimensional q-deformed double sinh-Gordon equation. The dynamical system approach is employed to study the model problem, from which we are able to obtain all possible bounded solutions, such as solitary wave solutions, kink and anti-kink wave solutions, and periodic wave solutions under different parameter conditions. Eleven exact parametric representations are provided explicitly.

Keywords:

Mathematics Subject Classification: 34C23, 35Q51, 35Q53, 58j55.

Citation:

Full Text(HTML)

Figure 1. Bifurcations of phase portraits of system (3) with $ (\alpha, \beta) = (-7,2.3) $

Download: Full-size image PowerPoint slide

Figure 2. Bifurcations of phase portraits of system (3) with $ (\alpha, \beta) = (7, -2.3) $

Download: Full-size image PowerPoint slide

Figure 3. Profile graph and 3d diagram of periodic wave

Download: Full-size image PowerPoint slide

Figure 4. Profile graph and 3d diagram of solitary wave

Download: Full-size image PowerPoint slide

Figure 5. Profile graph and 3d diagram of kink wave

Download: Full-size image PowerPoint slide

Figure 6. Profile graph and 3d diagram of anti-kink wave

Download: Full-size image PowerPoint slide

Figure 7. Profile graph and 3d diagram of solitary wave

Download: Full-size image PowerPoint slide

Figure 8. Profile graph and 3d diagram of periodic wave

Download: Full-size image PowerPoint slide

Figure 9. Profile graph and 3d diagram of periodic wave

Download: Full-size image PowerPoint slide

Figure 10. Profile graph and 3d diagram of solitary wave

Download: Full-size image PowerPoint slide

Figure 11. Changes of the level curves of system (5) as $ h $ varies in Fig. 2 (c)

Download: Full-size image PowerPoint slide

Figure 12. Profile graph and 3d diagram of periodic wave

Download: Full-size image PowerPoint slide

Figure 13. Profile graph and 3d diagram of solitary wave

Download: Full-size image PowerPoint slide

Figure 14. Profile graph and 3d diagram of solitary wave

Download: Full-size image PowerPoint slide

Figure 15. Profile graph and 3d diagram of solitary wave

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	A. Arab, Exactly solvable supersymmetric quantum mechanics, J. Math. Anal. Appl., 158 (1991), 63-79. doi: 10.1016/0022-247X(91)90267-4.
[2]	A. Arab, Exact solutions of multi-component nonlinear Schrödinger and Klein-Gordon equations in two-dimensional space-time, J. Phys. A: Math. Gen., 34 (2001), 4281-4288. doi: 10.1088/0305-4470/34/20/302.
[3]	A. Bastianello, B. Doyon, G. Watts, et al., Generalized hydrodynamics of classical integrable field theory: The sinh-Gordon model, SciPost Phys., 4 (2018), 045. doi: 10.21468/SciPostPhys.4.6.045.
[4]	P. F. Byrd and M. D. Fridman, Handbook of Elliptic Integrals for Engineers and Sciensists, Springer-Verlag, New York-Heidelberg, 1971.
[5]	I. Cabrera-Carnero and M. Moriconi, Noncommutative integrable field theories in 2d, Nucl. Phys. B, 673 (2003), 437-454. doi: 10.1016/j.nuclphysb.2003.09.014.
[6]	H. Eleuch, Some analytical solitary wave solutions for the generalized q-deformed Sinh-Gordon equation: $\frac{\partial^2\theta}{\partial z\partial\xi} = \alpha[\sinh_q(\beta\theta^{\gamma})]^p-\delta$, Adv. Math. Phys., 2018 (2018), Art. ID 5242757, 7 pp. doi: 10.1155/2018/5242757.
[7]	Y. Geng, T. He and J. Li, Exact travelling wave solutions for the $(n+1$)-dimensional double sine- and sinh-Gordon equations, Appl. Math. Comput., 188, (2007), 1513–1534. doi: 10.1016/j.amc.2006.11.014.
[8]	K. K. Kobayashi and M. Izutsu, Exact solution of $n-$dimensional sine-Gordon equation, J. Phys. Soc. Japan, 41 (1976), 1091-1092. doi: 10.1143/JPSJ.41.1091.
[9]	J. Li, Singular Nonlinear Traveling Wave Equations Bifurcations and Exact Solutions, Science Press, Beijing, 2013.
[10]	J. Li and G. Chen, On a class of singular nonlinear traveling wave equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17 (2007), 4049-4065. doi: 10.1142/S0218127407019858.
[11]	J. Li, W. Zhu and G. Chen, Understanding peakons, periodic peakons and compactons via a shallow water wave equation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650207, 27 pp. doi: 10.1142/S0218127416502072.
[12]	S. Liu, Z. Fu and S. Liu, Exact solutions to sine-Gordon-type equations, Phys Lett A., 351 (2006), 59-63. doi: 10.1016/j.physleta.2005.10.054.
[13]	K. Narita, Deformed sine- and sinh-Gordon equations, deformed Liuville equation, and their discrete models, J. Phys. Soc. Japan, 72 (2003), 1339-1349. doi: 10.1143/JPSJ.72.1339.
[14]	J. Song and J. Li, Bifurcations and exact travelling wave solutions for a shallow water wave model with a non-stationary bottom surface, J. Appl. Anal. Comput., 10 (2020), 350-360. doi: 10.11948/20190254.
[15]	N. K. Vitanov, On travelling waves and double-periodic structures in two-dimensional sine-Gordon systems, J. Physics. A. Math. Gen., 29 (1996), 5195-5207. doi: 10.1088/0305-4470/29/16/036.
[16]	N. K. Vitanov, Breather and soliton wave families for the sine-Gordon equation, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 454 (1998), 2409-2423. doi: 10.1098/rspa.1998.0264.
[17]	A. M. Wazwaz, The integrable time-dependent sine-Gordon equation with multiple optical kink solutions, Optik, 182 (2019), 605-610.