Exact solutions of a variant Boussinesq system

We apply the symmetry method based on the Frechet derivative of the differential operators to deduce the Lie symmetries of the following variant of the Boussinesq equations u(t) + alpha(1) (t)v(x) + beta(1) (t)uu(x) + gamma(1) (t)u(xx) = 0 v(t) + alpha(2) (t)uv(x) + beta 2(t)uv(x) + gamma(2)(t)v(xx) +p(t)u(xxx) = 0 where alpha(i)(t), beta(i)(t), gamma(i)(t), i = 1, 2 and p(t) are arbitrary functions of t. For each infinitesimal generator in the optimal system of subalgebras we study the reduced ODE and, among other solutions, furnish some nontrivial exact solutions in terms of hyperbolic functions. (c) 2006 Elsevier Ltd. All rights reserved.