The traveling wave solutions of the perturbed double Sine-Gordon equation

1. Faddeev, L., Popov, V.: Feynman diagrams for the Yang-Mills field. Phys. Lett. B 25(1), 29–30 (1967)

   Article  Google Scholar 

2. Rubinstein, J.: Sine-Gordon equation. J. Math. Phys. 11(1), 258–266 (1970)

   Article  MathSciNet  Google Scholar 

3. Ablowitz, M., Kaup, D., Newell, A., Segur, H.: Method for solving the Sine-Gordon equation. Phys. Rev. Lett. 30(25), 1262–1264 (1973)

   Article  MathSciNet  Google Scholar 

4. Guo, B., Pascual, P., Rodriguez, M., Vázquez, L.: Numerical solution of the Sine-Gordon equation. Appl. Math. Computat. 18(1), 1–14 (1986)

   Article  MathSciNet  Google Scholar 

5. Wazwaz, A.: The tanh method: exact solutions of the Sine-Gordon and the Sinh-Gordon equations. Appl. Math. Computat. 167(2), 1196–1210 (2005)

   Article  MathSciNet  Google Scholar 

6. Li, R., Geng, X.: Rogue periodic waves of the Sine-Gordon equation. Appl. Math. Lett. 102, 106147 (2020)

   Article  MathSciNet  Google Scholar 

7. 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)

   Article  MathSciNet  Google Scholar 

8. Alzaleq, L., Manoranjan, V.: Analytical solutions for the generalized sine-Gordon equation with variable coefficients. Phys. Scr. 96(5), 055218 (2021)

   Article  Google Scholar 

9. Dutykh, D., Caputo, J.: Wave dynamics on networks: method and application to the sine-Gordon equation. Appl. Numer. Math. 131, 54–71 (2018)

   Article  MathSciNet  Google Scholar 

10. Li, J., Chen, Y.: A physics-constrained deep residual network for solving the sine-Gordon equation. Commun. Theoret. Phys. 73(1), 015001 (2020)

    Article  MathSciNet  Google Scholar 

11. 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)

    Article  MathSciNet  Google Scholar 

12. 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)

    MathSciNet  Google Scholar 

13. Wang, M., Li, X.: Exact solutions to the double Sine-Gordon equation. Chaos, Solit. Fractals 27(2), 477–486 (2006)

    Article  MathSciNet  Google Scholar 

14. Salerno, M., Quintero, N.: Soliton ratchets. Phys. Rev. E 65(2), 025602 (2002)

    Article  MathSciNet  Google Scholar 

15. 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)

    Article  Google Scholar 

16. Deresse, A.: Double Sumudu transform iterative method for one-dimensional nonlinear coupled Sine-Gordon equation. Adv. Math. Phys. 2022, 1–15 (2022)

    MathSciNet  Google Scholar 

17. Bruce, A.: Is the \({\mathbb{Z} }_{2}\times {\mathbb{Z} }_{2}\)-graded Sine-Gordon equation integrable? Nuclear Phys. B 971, 115514 (2021)

    Article  Google Scholar 

18. Khusnutdinova, K., Pelinovsky, D.: On the exchange of energy in coupled Klein-Gordon equations. Wave Motion 38(1), 1–10 (2003)

    Article  MathSciNet  Google Scholar 

19. Denzler, J.: Nonpersistence of breather families for the perturbed Sine Gordon equation. Commun. Math. Phys. 158(2), 397–430 (1993)

    Article  MathSciNet  Google Scholar 

20. 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)

    Article  MathSciNet  Google Scholar 

21. McLaughlin, D., Scott, A.: Perturbation analysis of fluxon dynamics. Phys. Rev. A 18(4), 1652–1680 (1978)

    Article  Google Scholar 

22. 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)

    Article  MathSciNet  Google Scholar 

23. Holmes, P., Marsden, J.: Melnikov’s method and Arnold diffusion for perturbations of integrable Hamiltonian systems. J. Math. Phys. 23(4), 669–675 (1982)

    Article  MathSciNet  Google Scholar 

24. Lin, X.: Using Melnikov’s method to solve Silnikov’s problems. Proc. Royal Soc. Edinburgh Sect. A Math. 116(3–4), 295–325 (1990)

    Article  Google Scholar 

25. Yagasaki, K.: The method of Melnikov for perturbations of multi-degree-of-freedom Hamiltonian systems. Nonlinearity 12(4), 799–822 (1999)

    Article  MathSciNet  Google Scholar 

26. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)

    Article  MathSciNet  Google Scholar 

27. Fan, E.: Traveling wave solutions for nonlinear equations using symbolic computation. Comput. Math. Appl. 43(6–7), 671–680 (2002)

    Article  MathSciNet  Google Scholar 

28. 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)

    Article  MathSciNet  Google Scholar 

29. 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)

    Article  MathSciNet  Google Scholar 

30. Chicone, C.: The monotonicity of the period function for planar Hamiltonian vector fields. J. Differ. Equ. 69(3), 310–321 (1987)

    Article  MathSciNet  Google Scholar 

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