The sine–cosine and the tanh methods: Reliable tools for analytic treatment of nonlinear dispersive equations Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, 3700 West 103rd Street, Chicago, IL 60655, United States Abstract The sine–cosine method and the tanh method are used for analytic treatment of non- linear dispersive equations. Nonlinear variants of Boussinesq equation are used as vehi- cles to show the strength of these methods. Solutions of distinct physical structures: solitons, compactons, solitary patterns solutions and periodic solutions are formally derived. The results show that the change in physical structures of the obtained solu- tions depends mainly on exponents and on the coefficients of the derivatives involved. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Boussinesq equation; B(n, n) equation; Sine–cosine method; The tanh method; Non- linear dispersion; Compactons; Solitons 1. Introduction Nonlinear phenomena appear in many areas of scientific fields such as solid state physics, plasma physics, fluid dynamics, mathematical biology 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.02.047 E-mail address: wazwaz@sxu.edu Applied Mathematics and Computation 173 (2006) 150–164 www.elsevier.com/locate/amc