Abstract
In the paper, when \(0<c<1\), the phase portraits, traveling wave solutions and the minimum positive period of the periodic orbit for the double Sine-Gordon equation are discussed by using the dynamical system method and variable transformation. With the help of Melnikov’s function, the geometric singular perturbation theory and symbolic computation, we prove that the existence of traveling wave solution of the perturbed double Sine-Gordon equation for \(0<c<1\). The wave speed \(c=c(\alpha , \beta , \epsilon )\) and the constraints of parameters \(\alpha \), \(\beta \) are given.
Similar content being viewed by others
Data Availability
The data generated during the current study are available from the corresponding author on reasonable request.
Code availability
The codes generated during the current study are available from the corresponding author on reasonable request.
References
Faddeev, L., Popov, V.: Feynman diagrams for the Yang-Mills field. Phys. Lett. B 25(1), 29–30 (1967)
Rubinstein, J.: Sine-Gordon equation. J. Math. Phys. 11(1), 258–266 (1970)
Ablowitz, M., Kaup, D., Newell, A., Segur, H.: Method for solving the Sine-Gordon equation. Phys. Rev. Lett. 30(25), 1262–1264 (1973)
Guo, B., Pascual, P., Rodriguez, M., Vázquez, L.: Numerical solution of the Sine-Gordon equation. Appl. Math. Computat. 18(1), 1–14 (1986)
Wazwaz, A.: The tanh method: exact solutions of the Sine-Gordon and the Sinh-Gordon equations. Appl. Math. Computat. 167(2), 1196–1210 (2005)
Li, R., Geng, X.: Rogue periodic waves of the Sine-Gordon equation. Appl. Math. Lett. 102, 106147 (2020)
Porubov, A.V., Fradkov, A.L., Bondarenkov, R.S., Andrievsky, B.R.: Localization of the sine-Gordon equation solutions. Commun. Nonlinear Sci. Numer. Simulat. 39, 29–37 (2016)
Alzaleq, L., Manoranjan, V.: Analytical solutions for the generalized sine-Gordon equation with variable coefficients. Phys. Scr. 96(5), 055218 (2021)
Dutykh, D., Caputo, J.: Wave dynamics on networks: method and application to the sine-Gordon equation. Appl. Numer. Math. 131, 54–71 (2018)
Li, J., Chen, Y.: A physics-constrained deep residual network for solving the sine-Gordon equation. Commun. Theoret. Phys. 73(1), 015001 (2020)
Pelinovsky, D., White, R.: Localized structures on librational and rotational travelling waves in the sine-Gordon equation. Proc. Royal Soc. A 476(2242), 20200490 (2020)
Ige, Q., Oderinu, R.: Adomian polynomial and Elzaki transform method for solving Sine-Gordon equations. IAENG Int. J. Appl. Math. 49(3), 1–7 (2019)
Wang, M., Li, X.: Exact solutions to the double Sine-Gordon equation. Chaos, Solit. Fractals 27(2), 477–486 (2006)
Salerno, M., Quintero, N.: Soliton ratchets. Phys. Rev. E 65(2), 025602 (2002)
Rezazadeh, H., Zabihi, A., Davodi, A., Ansari, R., Ahmad, H., Yao, S.: New optical solitons of double Sine-Gordon equation using exact solutions methods. Results Phys. 49, 106452 (2023)
Deresse, A.: Double Sumudu transform iterative method for one-dimensional nonlinear coupled Sine-Gordon equation. Adv. Math. Phys. 2022, 1–15 (2022)
Bruce, A.: Is the \({\mathbb{Z} }_{2}\times {\mathbb{Z} }_{2}\)-graded Sine-Gordon equation integrable? Nuclear Phys. B 971, 115514 (2021)
Khusnutdinova, K., Pelinovsky, D.: On the exchange of energy in coupled Klein-Gordon equations. Wave Motion 38(1), 1–10 (2003)
Denzler, J.: Nonpersistence of breather families for the perturbed Sine Gordon equation. Commun. Math. Phys. 158(2), 397–430 (1993)
Maksimov, A., Pedersen, N., Christiansen, P., Molkov, J., Nekorkin, V.: On kink-dynamics of the perturbed Sine-Gordon equation. Wave Motion 23(2), 203–213 (1996)
McLaughlin, D., Scott, A.: Perturbation analysis of fluxon dynamics. Phys. Rev. A 18(4), 1652–1680 (1978)
Zhang, H., Xia, Y.: Persistence of kink and anti-kink wave solutions for the perturbed double Sine-Gordon equation. Appl. Math. Lett. 141, 108616 (2023)
Holmes, P., Marsden, J.: Melnikov’s method and Arnold diffusion for perturbations of integrable Hamiltonian systems. J. Math. Phys. 23(4), 669–675 (1982)
Lin, X.: Using Melnikov’s method to solve Silnikov’s problems. Proc. Royal Soc. Edinburgh Sect. A Math. 116(3–4), 295–325 (1990)
Yagasaki, K.: The method of Melnikov for perturbations of multi-degree-of-freedom Hamiltonian systems. Nonlinearity 12(4), 799–822 (1999)
Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)
Fan, E.: Traveling wave solutions for nonlinear equations using symbolic computation. Comput. Math. Appl. 43(6–7), 671–680 (2002)
Gao, X., Guo, Y., Shan, W.: Water-wave symbolic computation for the earth, enceladus and titan: the higher-order boussinesq-burgers system, auto-and non-auto-bäcklund transformations. Appl. Math. Lett. 104, 106170 (2020)
Li, J., Chen, G.: Bifurcations of traveling wave solutions for four classes of nonlinear wave equations. Int. J. Bifurcat. Chaos 15(12), 3973–3998 (2005)
Chicone, C.: The monotonicity of the period function for planar Hamiltonian vector fields. J. Differ. Equ. 69(3), 310–321 (1987)
Funding
The paper partly funded by the Special Projects in Key Areas of Colleges and Universities in Guangdong Province (No. 2020ZDZX3046), the Project of Guangdong University Engineering Technology Center (2022GCZX013).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declare that they have no Conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Yang, D. The traveling wave solutions of the perturbed double Sine-Gordon equation. J. Appl. Math. Comput. 70, 2241–2253 (2024). https://doi.org/10.1007/s12190-024-02048-w
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-024-02048-w