Wazwaz, Abdul-Majid New solitary wave solutions to the Kuramoto-Sivashinsky and the Kawahara equations. (English) Zbl 1107.65094 Appl. Math. Comput. 182, No. 2, 1642-1650 (2006). Summary: The Kuramoto-Sivashinsky equation and the Kawahara equation are studied. The tanh method and the extended tanh method are used for analytic treatment for these two equations. By means of these methods, new solitary wave solutions are determined for each equation. The two approaches are reliable and manageable. Cited in 41 Documents MSC: 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35Q51 Soliton equations 35Q53 KdV equations (Korteweg-de Vries equations) Keywords:Kuramoto-Sivashinsky equation; Kawahara equation; tanh method; solitons; periodic wave solutions Software:PDESpecialSolutions PDFBibTeX XMLCite \textit{A.-M. Wazwaz}, Appl. Math. Comput. 182, No. 2, 1642--1650 (2006; Zbl 1107.65094) Full Text: DOI References: [1] Conte, R., Exact solutions of nonlinear partial differential equations by singularity analysis, (Lecture Notes in Physics (2003), Springer), 1-83 · Zbl 1060.35001 [2] J. Rademacher, R. Wattenberg, Viscous shocks in the destabilized Kuramoto-Sivashinsky, J. Comput. Nonlin. Dynam., in press.; J. Rademacher, R. Wattenberg, Viscous shocks in the destabilized Kuramoto-Sivashinsky, J. Comput. Nonlin. Dynam., in press. [3] Wang, D.; Zhang, H.-Q., Further improved \(F\)-expansion method and new exact solutions of Nizhnik-Novikov-Veselov equation, Chaos Solitons Fractals, 25, 601-610 (2005) · Zbl 1083.35122 [4] Zhang, J-F.; Zheng, C-L, Abundant localized coherent structures of the (2+1)-dimensional generalized Nizhnik-Novikov-Veselov system, Chin. J. Phys., 41, 3, 242-254 (2003) [5] Bridges, T.; Derks, G., Linear instability of solitary wave solutions of the Kawahara equation and its generalizations, SIAM J. Math. Anal., 33, 6, 1356-1378 (2002) · Zbl 1011.35117 [6] Berloff, N. G.; Howard, L. N., Solitary and periodic solutions to nonlinear nonintegrable equations, Stud. Appl. Math., 99, 1-24 (1997) · Zbl 0880.35105 [7] Rosenau, P.; Pikovsky, A., Phase compactons in chains of dispersively coupled oscillations, Phys. Rev. Lett., 94, 174102, 14 (2005) [8] 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, 669705 (2004) · Zbl 1137.35324 [9] Goktas, U.; Hereman, W., Symbolic computation of conserved densities for systems of nonlinear evolution equations, J. Symbolic Comput., 24, 591621 (1997) [10] Wazwaz, A. M., The tanh method for travelling wave solutions of nonlinear equations, Appl. Math. Comput., 154, 3, 713723 (2004) [11] Wazwaz, A. M., Partial Differential Equations: Methods and Applications (2002), Balkema Publishers: Balkema Publishers The Netherlands · Zbl 0997.35083 [12] Wazwaz, A. M., Compactons in a class of nonlinear dispersive equations, Math. Comput. Modell., 37, 3/4, 333341 (2003) [13] Wazwaz, A. M., Distinct variants of the NNVV equation with compact and noncompact structures, Appl. Math. Comput., 150, 365377 (2004) [14] Wazwaz, A. M., Variants of the generalized NNVV equation with compact and noncompact structures, Comput. Math. Appl., 47, 583591 (2004) [15] Malfliet, W.; Hereman, W., The tanh method: I. Exact solutions of nonlinear evolution and wave equations, Phys. Scripta, 54, 563568 (1996) [16] Malfliet, W.; Hereman, W., The tanh method: II. Perturbation technique for conservative systems, Phys. Scripta, 54, 569575 (1996) 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.