The tanh and the sine–cosine methods for compact and noncompact solutions of the nonlinear Klein–Gordon equation Abdul-Majid Wazwaz Department of Mathematics and Computer Science, Saint Xavier University, Chicago, IL 60655, USA Abstract The nonlinear Klein–Gordon equation is used as a vehicle to employ the tanh method and the sine–cosine method to formally derive a number of travelling wave solu- tions. The study features a variety of solutions with distinct physical structures. The work shows that one method complements the other, and each method gives solutions of formal properties. The obtained solutions include compactons, solitons, solitary pat- terns, and periodic solutions. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Compactons; Solitons; Periodic solutions; Klein–Gordon equation; Sine–cosine method; The tanh method 1. Introduction The quasilinear Klein–Gordon equation u tt  au xx þ bu  ku 3 ¼ 0; ð1Þ 0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.08.006 E-mail address: wazwaz@sxu.edu Applied Mathematics and Computation 167 (2005) 1179–1195 www.elsevier.com/locate/amc