Exact solutions of the complex modified Korteweg-de Vries equation

Summary: Firstly, similarity reductions of the complex modified Korteweg-de Vries equation (CMKdV), which arises in the asymptotic interpretation of one- dimensional plane-wave propogation in a quadratic micropolar medium are discussed. Although it is not a soliton equation solvable by inverse scattering transformation, its similarity reductions obtained by the use of Lie group methods are of mathematical interest. Secondly, the Painlevé analysis developed by Weiss et al for nonlinear partial differential equations is applied to the CMKdV equation, and the data obtained by the truncation technique yield some analytical solutions of the ordinary modified Korteweg-de Vries equation and travelling-wave solutions of the CMKdV equation which are also solutions of the similarity reduction obtained by classical Lie group analysis.