Classical and nonclassical symmetries of a generalized Boussinesq equation.

*(English)*Zbl 0944.35084Summary: We apply the Lie-group formalism and the nonclassical method due to Bluman and Cole to deduce symmetries of the generalized Boussinesq equation

$${u}_{tt}-{u}_{xx}+{\left(f\left(u\right)+{u}_{xx}\right)}_{xx}=0,$$

which has the classical Boussinesq equation as an special case. We study the class of functions $f\left(u\right)$ for which this equation admits either the classical or the nonclassical method. Symmetry reductions are obtained and some new exact solutions are derived.

##### MSC:

35Q53 | KdV-like (Korteweg-de Vries) equations |

37K30 | Relations of infinite-dimensional systems with algebraic structures |

35A30 | Geometric theory for PDE, characteristics, transformations |