# zbMATH — the first resource for mathematics

The tanh-coth method for solitons and kink solutions for nonlinear parabolic equations. (English) Zbl 1119.65100
Summary: The tanh-coth method is used to derive solitons and kink solutions for some of the well-known nonlinear parabolic partial differential equations. The equations include the Fisher equation, Newell-Whithead equation, Allen-Cahn equation, Fitzhugh-Nagumo equation, and the Burgers-Fisher equation. The new tanh-coth approach provides abundant solitons and kink solutions in addition to the existing ones. The power of this manageable method is confirmed.

##### MSC:
 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35Q53 KdV equations (Korteweg-de Vries equations) 35Q51 Soliton equations
Full Text:
##### References:
 [1] Polyanin, A.D.; Zaitsev, V.F., Handbook of nonlinear partial differential equations, (2004), Chapman & Hall/CRC · Zbl 1024.35001 [2] Voigt, A., Asymptotic behavior of solutions to the allen – cahn equation in spherically symmetric domains, Caesar preprints, 1-8, (2001) [3] Allen, S.M.; Cahn, J.W., A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening, Acta metall., 27, 1085-1095, (1979) [4] Rosu, H.C.; Cornejo-Pérez, O., Supersymmetric pairing of kinks for polynomial nonlinearities, Phys. rev. E, 71, 1-13, (2005) [5] Wazwaz, A.M.; Gorguis, A., An analytic study of fisher’s equation by using Adomian decomposition method, Appl. math. comput., 154, 3, 609-620, (2004) · Zbl 1054.65107 [6] Hirota, R., Direct methods in soliton theory, (1980), Springer Berlin [7] Ablowitz, M.; Segur, H., Solitons and the inverse scattering transform, (1981), SIAM Philadelphia · Zbl 0472.35002 [8] Lax, P.D., Integrals of nonlinear equations of evolution and solitary waves, Comm. pure appl. math., 62, 467-490, (1968) · Zbl 0162.41103 [9] Malfliet, W., Solitary wave solutions of nonlinear wave equations, Am. J. phys., 60, 7, 650-654, (1992) · Zbl 1219.35246 [10] Malfliet, W.; Hereman, W., The tanh method: I. exact solutions of nonlinear evolution and wave equations, Phys. scripta, 54, 563-568, (1996) · Zbl 0942.35034 [11] Malfliet, W.; Hereman, W., The tanh method: II. perturbation technique for conservative systems, Phys. scripta, 54, 569-575, (1996) · Zbl 0942.35035 [12] Goktas, U.; Hereman, W., Symbolic computation of conserved densities for systems of nonlinear evolution equations, J. symbolic comput., 24, 591-621, (1997) · Zbl 0891.65129 [13] Baldwin, D.; Goktas, U.; Hereman, W.; Hong, L.; Martino, R.S.; Miller, J.C., Symbolic computation of exact solutions in hyperbolic and elliptic functions for nonlinear pdes, J. symbolic comput., 37, 669-705, (2004) · Zbl 1137.35324 [14] Hereman, W.; Nuseir, A., Symbolic methods to construct exact solutions of nonlinear partial differential equations, Math. comput. simul., 43, 13-27, (1997) · Zbl 0866.65063 [15] Wazwaz, A.M., The tanh method for travelling wave solutions of nonlinear equations, Appl. math. comput., 154, 3, 713-723, (2004) · Zbl 1054.65106 [16] Wazwaz, A.M., Partial differential equations: methods and applications, (2002), Balkema Publishers The Netherlands · Zbl 0997.35083 [17] Wazwaz, A.M., The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations, Appl. math. comput., 49, 565-574, (2005) · Zbl 1070.35043 [18] Wazwaz, A.M., Variants of the two-dimensional Boussinesq equations with compactons, solitons and periodic solutions, Comput. math. appl., 49, 295-301, (2005) · Zbl 1070.35042 [19] Wazwaz, A.M., The tanh and the sine – cosine methods for a reliable treatment of the modified equal width equation and its variants, Commun. nonlinear sci. numer. simul., 112, 148-160, (2006) · Zbl 1078.35108 [20] Wazwaz, A.M., New compactons, solitons and periodic solutions for nonlinear variants of the KdV and the KP equations, Chaos, solitons compactons, 22, 1, 249-260, (2004) · Zbl 1062.35121 [21] Wazwaz, A.M., The tanh method for generalized forms of nonlinear heat conduction and burgers – fisher equation, Appl. math. comput., 169, 321-338, (2005) · Zbl 1121.65359 [22] A.M. Wazwaz, New solitary wave solutions to the modified forms of Degasperis-Procesi and Camassa-Holm equations, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.07.092. · Zbl 1114.65124 [23] A.M. Wazwaz, The extended tanh method for new solitons solutions for many forms of the fifth-order KdV equations, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.07.002. · Zbl 1115.65106 [24] A.M. Wazwaz, The extended tanh method for abundant solitary wave solutions of nonlinear wave equations, Appl. Math. Comput., in press, doi:10.1016/j.amc.2006.09.013. · Zbl 1118.65370
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.