Bifurcations of travelling wave solutions in the \((N + 1)\)-dimensional sine-cosine-Gordon equations. (English) Zbl 1222.35023
Summary: The \((N + 1)\)-dimensional sine–cosine-Gordon equations are studied. The existence of solitary wave, kink and anti-kink wave, and periodic wave solutions are proved, by using the method of bifurcation theory of dynamical systems. All possible bounded exact explicit parametric representations of the above travelling solutions are obtained.
MSC:
| 35B32 | Bifurcations in context of PDEs |
| 35C07 | Traveling wave solutions |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
solitary travelling wave solution; kink and anti-kink travelling wave solution; periodic travelling wave solution; \((N + 1)\)-dimensional sine–cosine-Gordon equationsReferences:
| [1] | Wang, Deng-Shan; Yan, Zhenya; Li, Hongbo, Some special types of solutions of a class of the \((N+1)\)-dimensional nonlinear wave equations, Comput Math Appl, 56, 1569-1579 (2008) · Zbl 1155.35444 |
| [2] | Chow, S. N.; Hale, J. K., Method of bifurcation theory (1981), Springer-Verlag: Springer-Verlag New York |
| [3] | Geng, Yixiang; He, Tianlan; Li, Jibin, Exact travelling wave solutions for the \((n+1)\)-dimensional double sine-and sinh-Gordon equations, Appl Math Comput, 188, 1513-1534 (2007) · Zbl 1118.35042 |
| [4] | Li, Jibin; Liu, Zhenglong, Smooth and non-smooth travelling waves in a nonlinear dispersive equation, Appl Math Model, 25, 41-51 (2000) · Zbl 0985.37072 |
| [5] | Kobayashi, K. K.; Izutsu, M., Exact solution of \(n\)-dimensional sine-Gordon equation, J Phys Soc Jpn, 41, 3, 1091-1092 (1976) · Zbl 1334.35284 |
| [6] | Perring, J. K.; Skyrme, T. H., A model unified field equation, Nucl Phys, 31, 550-555 (1962) · Zbl 0106.20105 |
| [7] | Wazwaz, A. M., The tanh method: exact solutions of the Sine-Gordon and Sinh-Gordon equations, Appl Math Comput, 167, 1196-1210 (2005) · Zbl 1082.65585 |
| [8] | Wazwaz, A. M., Exact solutions for the generalized sine-Gordon and the generalized sinh-Gordon equations, Chaos Solitons Fract, 28, 127-135 (2006) · Zbl 1088.35544 |
| [9] | Wazwaz, A. M., Travelling wave solutions for combined and double combined sine-cosine-Gordon equations by the variable separated ODE method, Appl Math Comput, 177, 755-760 (2006) · Zbl 1099.65095 |
| [10] | Wazwaz, A. M., The tanh method and a variable separated ODE method for solving double sine-Gordon equation, Phys Lett A, 350, 367-370 (2006) · Zbl 1195.65210 |
| [11] | Fabian, Anne L.; Kohl, Russell; Biswas, Anjan, Perturbation of topological solitons due to sine-Gordon equation and its type, Commun Nonlinear Sci Numer Simul, 14, 1227-1244 (2009) · Zbl 1221.37121 |
| [12] | Byrd, P. F.; Fridman, M. D., Handbook of elliptic integrals for engineers and sciensists (1971), Springer: Springer Berlin · Zbl 0213.16602 |
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