Group classification of nonlinear evolution equations: Semi-simple groups of contact transformations Qing Huang a,⇑ , Renat Zhdanov b a Department of Mathematics, Northwest University, Xi’an 710069, China b BIO-key International, 55121 Eagan, MN, USA article info Article history: Received 25 March 2014 Received in revised form 16 December 2014 Accepted 19 January 2015 Available online 31 January 2015 Keywords: Group classification Semi-simple group Contact transformation abstract We generalize and modify the group classification approach of Zhdanov and Lahno (1999) to make it applicable beyond Lie point symmetries. This approach enables obtaining exhaustive classification of second-order nonlinear evolution equations in one spatial dimension invariant under semi-simple groups of contact transformations. What is more, all inequivalent second-order nonlinear evolution equations which admit semi-simple groups or groups having nontrivial Levi decompositions are constructed in explicit forms. Ó 2015 Elsevier B.V. All rights reserved. 1. Introduction In this paper, we study contact symmetries of general second-order nonlinear evolution equations of the form u t ¼ F ðt; x; u; u 1 ; u 2 Þ; ð1Þ where u ¼ uðt; xÞ; u t ¼ @u=@t; u i ¼ @ i u=@x i (i 2 NÞ, and F is an arbitrary sufficiently smooth real-valued function with F u 2 – 0. In fact, we intend to construct all possible forms of the function F such that the corresponding equation admits nontrivial contact transformation group which contains a semi-simple subgroup. The class (1) includes a number of important and fundamental equations of modern mathematical and theoretical physics, such as the heat, Fisher, Newell–Whitehead and Burgers equations, to mention just a few (see, e.g., [17,22,31]). Lie group analysis is universally recognized as a versatile and convenient tool for analysis of partial differential equations (PDEs). However, there is a necessary prerequisite for efficient utilization of any group-theoretical method. Namely, the equation under study has to admit a nontrivial Lie group. By this very reason, the problem of group classification of nonlinear PDEs has attracted so much attention and resulted in numerous publications recently. In the case when a transformation involves dependent and independent variables only, it is called point transformation. For the more general case of transformation including first derivatives of the dependent variables, the term contact transformation has been adopted in the literature. Nowadays, the point group classification of the class (1) has been extensively studied (see [32,3,33] and the references therein). In contrast, much less attention has been devoted to the contact symmetries of the class (1). The notion of contact (tangential) transformation within the context of differential equations (DEs) was first presented in Sophus Lie’s doctoral thesis [15]. He obtained a number of classical results on contact symmetries of ordinary differential http://dx.doi.org/10.1016/j.cnsns.2015.01.009 1007-5704/Ó 2015 Elsevier B.V. All rights reserved. ⇑ Corresponding author. E-mail address: hqing@nwu.edu.cn (Q. Huang). Commun Nonlinear Sci Numer Simulat 26 (2015) 184–194 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns