SYMMETRY REDUCTIONS, EXACT-SOLUTIONS, AND PAINLEVE ANALYSIS FOR A GENERALIZED BOUSSINESQ EQUATION

In this paper, new nonclassical symmetry reductions and exact solution s are presented fora Generalised Boussinesq equation u(xxxx) + pu(t)u( xx) + qu(x)u(xt) + ru(x)(2)u(xx) + u(u) = O, which has the modified Bo ussinesq equation (q = O, r = -1/2p(2)) and dispersive water wave equa tion, or classical Boussinesq equations (q = 2p, r = 3/2p(2)) as speci al cases. These symmetry reductions are obtained using the Direct Meth od, originally developed by Clarkson and Kruskal to study symmetry red uctions of the Boussinesq equation, which involves no group theoretic techniques, and using these reductions, we obtain exact solutions expr essible in terms of solutions of the second and fourth Painleve equati ons, Jacobi and Weierstrass elliptic functions, and elementary functio ns, for certain values of the parameters p, q, and r. Furthermore, in the case when q = p and r = 1/2p(2), symmetry reductions are obtained which are reminiscent of reductions of the 2 + 1-dimensional cubic non linear Schrodinger equation arising from the Talanov lens transformati on. (C) 1994 Academic Press, Inc.