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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.

65M70Spectral, collocation and related methods (IVP of PDE)
35Q53KdV-like (Korteweg-de Vries) equations
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
Full Text: DOI
[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