Painlevé analysis, Lie symmetries and exact solutions for (2+1)-dimensional variable coefficients Broer–Kaup equations Sachin Kumar a , K. Singh b , R.K. Gupta a,⇑ a School of Mathematics and Computer Applications, Thapar University, Patiala 147004, Punjab, India b Department of Mathematics, Jaypee University of Information Technology, Waknaghat, P.O. Dumehar Bani, Kandaghat, Distt. Solan, 173215 H.P., India article info Article history: Received 11 June 2011 Received in revised form 20 August 2011 Accepted 1 September 2011 Available online 13 September 2011 Keywords: (2+1)-Variable coefficients Broer–Kaup (VCBK) equations Painlevé analysis Lie classical method Exact solutions abstract A (2+1) dimensional Broer–Kaup system which is obtained from the constraints of the KP equation is of importance in mathematical physics field. In this paper, the Painlevé analysis of (2+1)-variable coefficients Broer–Kaup (VCBK) equation is performed by the Weiss– Kruskal approach to check the Painlevé property. Similarity reductions of the VCBK equa- tion to one-dimensional partial differential equations including Burger’s equation are investigated by the Lie classical method. The Lie group formalism is applied again on one of the investigated partial differential equation to derive symmetries, and the ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction A large variety of physical, chemical, and biological phenomena is governed by nonlinear evolution equations. Many com- pletely integrable models were presented during the course of studying shallow water waves, for example, KdV-type equa- tions, the WBK equation, the integrable long wave equation, the Boussinesq equation, etc. Finding exact solutions of these nonlinear evolution equations plays an important role for these equations which are drawn from diverse interesting nonlin- ear phenomena. As a result, the research on exact solutions of nonlinear evolution equations has become more and more important. A (2+1)-dimensional Broer–Kaup–Kupershmidt (BKK) system [1,2] H ty ¼ H xxy  2ðHH x Þ y  2G xx ; G t ¼G xx  2ðGHÞ x ; ð1:1Þ can be obtained from the symmetry reduction of KP equation [3]. It has been widely applied in many branches of physics like plasma physics, fluid dynamics, nonlinear fiber optic communication, etc. The BKK system was used to model nonlinear and dispersive long gravity waves traveling in two horizontal directions on shallow water of uniform depth, and can be derived from KP equation. The physical situations in which nonlinear equations arise tend to be highly idealized due to assumption of constant coefficients. Due to this, much attention has been paid on study of nonlinear equations with variable coefficients [4–8]. Often, it is very difficult to solve explicitly these nonlinear equations for exact solutions. Consequently, perturbation, asymptotic and numerical methods are applied to obtain approximate solutions of these equations. However, there is much 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.09.003 ⇑ Corresponding author. E-mail addresses: sachin1jan@yahoo.com (S. Kumar), karan_jeet@yahoo.com (K. Singh), rajeshgupta@thapar.edu (R.K. Gupta). Commun Nonlinear Sci Numer Simulat 17 (2012) 1529–1541 Contents lists available at SciVerse ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns