The symmetry reductions of a turbulence model

In this paper we obtain symmetry reductions of the system of two coupled parabolic partial differential equations which model the evolution of turbulent bursts using the classical Lie method of infinitesimals. The reduction to systems of ordinary differential equations (ODEs) are obtained from the optimal system of subalgebras. These systems admit symmetries which lead to further reductions. An algorithm presented by Bluman for reducing the order of ODEs allows us to reduce some of these systems, invariant under a two-parameter group, directly to first-order ODEs systems. The hidden symmetries of some of these systems are obtained and some new exact solutions have been derived.