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The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations. (English) Zbl 1082.65585
Summary: We use the tanh method for traveling wave solutions of the sine-Gordon and the sinh-Gordon equations. Several exact solutions of distinct physical structures are obtained. The method is powerful with minimal algebra work and is demonstrated for four models.

##### MSC:
 65M70 Spectral, collocation and related methods (IVP of PDE) 35Q53 KdV-like (Korteweg-de Vries) equations 37K10 Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
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##### References:
 [1] Perring, J. K.; Skyrme, T. H.: A model unified field equation. Nucl. phys. 31, 550-555 (1962) · Zbl 0106.20105 [2] Ablowitz, M. J.; Herbst, B. M.; Schober, C.: On the numerical solution of the sine-Gordon equation. J. comput. Phys. 126, 299-314 (1996) · Zbl 0866.65064 [3] Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, No. 7, 650-654 (1992) · Zbl 1219.35246 [4] W. Malfliet, The tanh method. I. Exact solutions of nonlinear evolution and wave equations, Phys. Scr. 54, 563-568. · Zbl 0942.35034 [5] W. Malfliet, The tanh method. II. Perturbation technique for conservative systems, Phys. Scr. 54, 569-575. · Zbl 0942.35035 [6] Khater, A. H.; Malfliet, W.; Callebaut, D. K.; Kamel, E. S.: The tanh method, a simple transformation and exact analytical solutions for nonlinear reaction-diffusion equations. Chaos, solitons fract. 14, 513-522 (2002) · Zbl 1002.35066 [7] Parkes, E. J.; Duffy, B. R.: An automated tanh-function method for finding solitary wave solutions to non-linear evolution equations. Comput. phys. Commun. 98, 288-300 (1996) · Zbl 0948.76595 [8] Glasner, K.: Nonlinear preconditioning for diffuse interfaces. J. comput. Phys. 174, 695-711 (2001) · Zbl 0991.65076 [9] Kawahara, T.; Tanaka, M.: Interactions of traveling fronts: an exact solution of a nonlinear diffusion equation. Phys. lett. A 97, No. 8, 311-314 (1983) [10] Conte, R.; Mussette, M.: Link between solitary waves and projective Riccati equations. J. phys.: math. Gen. 25, 5609-5623 (1992) · Zbl 0782.35065 [11] Huibin, L.; Kelin, W.: Exact solutions for two nonlinear equations: I. J. phys. A: math. Gen. 23, 3923-3928 (1990) · Zbl 0718.35020 [12] M. Wang, Exact solutions for a compound KdV-Burgers equation, Phys. Lett. A, 213, 279-287. · Zbl 0972.35526 [13] Ma, W.: Travelling wave solutions to a seventh order generalized KdV equation. Phys. lett. A 180, 221-224 (1993) [14] Wei, G. W.: Discrete singular convolution for the sine-Gordon equation. Physica D 137, 247-259 (2000) · Zbl 0944.35087 [15] Fan, E.; Hon, Y. C.: Generalized tanh method extended to special types of nonlinear equations. Z. naturforsch. A 57a, 692-700 (2002) [16] Wazwaz, A. M.: Partial differential equations: methods and applications. (2002) · Zbl 1079.35001 [17] Wazwaz, A. M.: New solitary-wave special solutions with compact support for the nonlinear dispersive $K(m,n)$ equations. Chaos, solitons fract. 13, No. 2, 321-330 (2002) · Zbl 1028.35131 [18] Wazwaz, A. M.: Exact specific solutions with solitary patterns for the nonlinear dispersive $K(m,n)$ equations. Chaos, solitons fract. 13, No. 1, 161-170 (2001) [19] Wazwaz, A. M.: General compactons solutions for the focusing branch of the nonlinear dispersive $K(n,n)$ equations in higher dimensional spaces. Appl. math. Comput. 133, No. 2/3, 213-227 (2002) · Zbl 1027.35117 [20] Wazwaz, A. M.: General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive $K(n,n)$ equations in higher dimensional spaces. Appl. math. Comput. 133, No. 2/3, 229-244 (2002) · Zbl 1027.35118 [21] Wazwaz, A. M.: A study of nonlinear dispersive equations with solitary-wave solutions having compact support. Math. comput. Simul. 56, 269-276 (2001) · Zbl 0999.65109 [22] Wazwaz, A. M.: Compactons dispersive structures for variants of the $K(n,n)$ and the KP equations. Chaos, solitons fract. 13, No. 5, 1053-1062 (2002) · Zbl 0997.35083 [23] Wazwaz, A. M.: Compactons and solitary patterns structures for variants of the KdV and the KP equations. Appl. math. Comput. 139, No. 1, 37-54 (2003) · Zbl 1029.35200 [24] Wazwaz, A. M.: Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method. Chaos, solitons fract. 12, No. 8, 1549-1556 (2001) · Zbl 1022.35051 [25] Wazwaz, A. M.: A computational approach to soliton solutions of the Kadomtsev-petviashili equation. Appl. math. Comput. 123, No. 2, 205-217 (2001) · Zbl 1024.65098 [26] 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