Group classification, optimal system and optimal reductions of a class of Klein Gordon equations H. Azad a , M.T. Mustafa a, * , M. Ziad b a Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia b Department of Mathematics and Statistics, College of Science, Sultan Qaboos University, Oman article info Article history: Received 13 May 2009 Accepted 16 May 2009 Available online 23 May 2009 PACS: 02.30.Jr 02.20.a Keywords: Nonlinear wave equation Lie symmetries Group classification Optimal system Invariant solutions abstract Complete symmetry analysis is presented for non-linear Klein Gordon equations u tt ¼ u xx þ f ðuÞ. A group classification is carried out by finding f ðuÞ that give larger symme- try algebra. One-dimensional optimal system is determined for symmetry algebras obtained through group classification. The subalgebras in one-dimensional optimal system and their conjugacy classes in the corresponding normalizers are employed to obtain, up to conjugacy, all reductions of equation by two-dimensional subalgebras. This is a new idea which improves the computational complexity involved in finding all possible reductions of a PDE of the form Fðx; t; u; u x ; u t ; u xx ; u tt ; u xt Þ¼ 0 to a first order ODE. Some exact solutions are also found. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction This paper gives a complete symmetry analysis of a class of non-linear wave equations. We follow ideas of Ovsiannikov [15], Ibragimov [6,7] and Clarkson and Mansfield [12] to carry this through. The equations considered are the non-linear Klein Gordon equations in one space dimension, namely the non-linear wave equations of the form u tt ¼ u xx þ f ðuÞ; ðf uu –0Þ ð1:1Þ The analysis consists of first finding the Lie symmetries of equation with arbitrary f ðuÞ and then determining all possible forms of f ðuÞ for which larger symmetry groups exist. This is followed by the determination of optimal systems of subalge- bras and reductions and invariant solutions corresponding to these optimal systems. The first group classification problem was carried out by Ovsiannikov [15] who classified all forms of the non-linear heat equation u t ¼ðf ðuÞu x Þ x . Studies related to group properties of non-linear wave equations began with the well-known paper of Ames [1] in 1981. The group classification problem for the equation u tt ¼ u xx þ u yy þ u zz þ f ðuÞ in three space dimensions has been carried out by Rudra in [16]. However, we have followed ideas of Clarkson and Mansfield [12] to carry this through because of the link with Groebner bases and their potential wider applicability. 1007-5704/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2009.05.045 * Corresponding author. E-mail addresses: hassanaz@kfupm.edu.sa (H. Azad), tmustafa@kfupm.edu.sa (M.T. Mustafa), mziad@squ.edu.om (M. Ziad). Commun Nonlinear Sci Numer Simulat 15 (2010) 1132–1147 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns