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Bifurcations of travelling wave solutions for the generalized double sinh-Gordon equation. (English) Zbl 1124.35077
Summary: The generalized double sinh-Gordon equation is studied. The existence of periodic wave, solitary wave, kink and anti-kink wave and unbounded wave solutions is proved, by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of the above waves are determined.

MSC:
35Q53KdV-like (Korteweg-de Vries) equations
37K50Bifurcation problems (infinite-dimensional systems)
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References:
[1] Chow, S. N.; Hale, J. K.: Method of bifurcation theory. (1981)
[2] Li, Jibin; Chen, Guanrong: Bifurcations of travelling wave solutions for four classes of nonlinear wave equations. Int. bifurcat. Chaos 15, 3973-3998 (2005) · Zbl 1093.35055
[3] Li, Jibin; Liu, Zhenrong: Smooth and non-smooth travelling waves in a nonlinearly dispersive equation. Appl. math. Model. 25, 41-56 (2000) · Zbl 0985.37072
[4] Li, Jibin; Liu, Zhenrong: Travelling wave solutions for a class of nonlinear dispersive equations. Chin. ann. Math. 23B, 397-418 (2002) · Zbl 1011.35014
[5] Wazwaz, A. M.: The variable separated ODE and the tanh methods for solving the combined and the double combined sinh -- cosh Gordon equations. Appl. math. Comput. 177, 745-754 (2005) · Zbl 1096.65104