Symmetry Reductions and Exact Solutions for a Generalised Boussinesq Equation

Abstract

In this paper, new nonclassical symmetry reductions and exact solutions are presented for a Generalised Boussinesq equation \( {u_{{xxxx}}} + p{u_t}{u_{{xx}}} + q{u_x}{u_{{xx}}} + ru_x^2{u_{{xx}}} + {u_{{tt}}} = 0 \), which has the modified Boussinesq equation (\( (q = 0,\,r = - \frac{1}{2}{p^2}) \)) and dispersive water wave equation, or classical Boussinesq equations (\( q = 2p,\,r = \frac{3}{2}{p^2} \)) as special cases. These symmetry reductions are obtained using the the Direct Method, originally developed by Clarkson & Kruskal to study symmetry reductions of the Boussinesq equation, which involves no group theoretic techniques and using these reductions, we obtain exact solutions expressible in terms of solutions of the second and fourth Painleve equations, Jacobi and Weierstrass elliptic functions, and elementary functions, for certain values of the parameters p, q and r. Furthermore, in the case when q = p and \( r = \frac{1}{2}{p^2} \), symmetry reductions are obtained which are reminiscent of reductions of the 2 + 1-dimensional cubic nonlinear Schrodinger equation arising from the Talanov lens transformation.