Group classification and exact solutions of a higher-order Boussinesq equation

We consider a family of sixth-order Boussinesq equations in one space dimension with an arbitrary nonlinearity. The equation was originally derived for one-dimensional lattice model considering higher order effects, and later it re-derived the context nonlinear nonlocal elasticity. In sense wave propagation solids, nonlinearity function is connected stress-strain relation physical model. Considering general nonlinearity, we determine classes so that certain type Lie symmetry algebra admitted this family. find maximal four, which realized when assumes some special canonical form. After perform reductions to ordinary differential equations. case quadratic provide several exact solutions, are terms elliptic functions.