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doi: 10.3934/dcdss.2022113
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## Bifurcations and exact traveling wave solutions of the (n+1)-dimensional q-deformed double sinh-Gordon equations

 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

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.

Citation: Jie Song. Bifurcations and exact traveling wave solutions of the (n+1)-dimensional q-deformed double sinh-Gordon equations. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022113
##### 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.

show all references

##### 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.
Bifurcations of phase portraits of system (3) with $(\alpha, \beta) = (-7,2.3)$
Bifurcations of phase portraits of system (3) with $(\alpha, \beta) = (7, -2.3)$
Profile graph and 3d diagram of periodic wave
Profile graph and 3d diagram of solitary wave
Profile graph and 3d diagram of kink wave
Profile graph and 3d diagram of anti-kink wave
Profile graph and 3d diagram of solitary wave
Profile graph and 3d diagram of periodic wave
Profile graph and 3d diagram of periodic wave
Profile graph and 3d diagram of solitary wave
Changes of the level curves of system (5) as $h$ varies in Fig. 2 (c)
Profile graph and 3d diagram of periodic wave
Profile graph and 3d diagram of solitary wave
Profile graph and 3d diagram of solitary wave
Profile graph and 3d diagram of solitary wave
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