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Exact solutions to the double sinh-Gordon equation by the tanh method and a variable separated ODE method. (English) Zbl 1089.35534
Summary: New exact travelling wave solutions for the double sinh-Gordon equation and its generalized form are formally derived by using the tanh method and the variable separated ODE method. The Painlevé property $v = e^{u}$ is employed to support the tanh method in deriving exact solutions. The work emphasizes the power of the methods in providing distinct solutions of different physical structures.

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35K40 Systems of second-order parabolic equations, general 35C05 Solutions of PDE in closed form
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##### References:
 [1] Sirendaoreji; Jiong, S.: A direct method for solving sinh-Gordon type equation. Phys. lett. A 298, 133-139 (2002) · Zbl 0995.35056 [2] Fu, Z.; Liu, S.; Liu, S.: Exact solutions to double and triple sinh-Gordon equations. Z. naturforsch. 59a, 933-937 (2004) [3] Perring, J. K.; Skyrme, T. H.: A model unified field equation. Nucl. phys. 31, 550-555 (1962) · Zbl 0106.20105 [4] Whitham, G. B.: Linear and nonlinear waves. (1999) · Zbl 0940.76002 [5] Infeld, E.; Rowlands, G.: Nonlinear waves, solitons and chaos. (2000) · Zbl 0994.76001 [6] Polyanin, A.; Zaitsev, V. F.: Handbook of nonlinear partial differential equations. (2004) · Zbl 1053.35001 [7] Ablowitz, M. J.; Herbst, B. M.; Schober, C.: On the numerical solution of the sinh-Gordon equation. Journal of computational physics 126, 299-314 (1996) · Zbl 0866.65064 [8] Wei, G. W.: Discrete singular convolution for the sinh-Gordon equation. Physica D 137, 247-259 (2000) · Zbl 0944.35087 [9] Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, No. 7, 650-654 (1992) · Zbl 1219.35246 [10] W. Malfliet, The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Physica Scripta 54, 563--568. · Zbl 0942.35034 [11] W. Malfliet, The tanh method: II. Perturbation technique for conservative systems, Physica Scripta 54, 569--575. · Zbl 0942.35035 [12] A.M. Wazwaz, The tanh method: Exact solutions of the sinh-Gordon and the Sinh-Gordon equations, Appl. Math. Comput. (to appear). · Zbl 1082.65585 [13] Wazwaz, A. M.: The tanh method for travelling wave solutions of nonlinear equations. Appl. math. Comput. 154, No. 3, 713-723 (2004) · Zbl 1054.65106 [14] Wazwaz, A. M.: Partial differential equations: methods and applications. (2002) · Zbl 1079.35001 [15] A.M. Wazwaz, The variable separated ODE and the tanh methods for solving the combined and the double combined sinh-cosh-Gordon equations, Appl. Math. Comput. (to appear). · Zbl 1096.65104 [16] A.M. Wazwaz, Travelling wave solutions for combined and double combined sine-cosine-Gordon equations, Appl. Math. Comput. (to appear). · Zbl 1099.65095