  A formally second order correct Boussinesq-type equation that describes
unidirectional shallow water waves is derived,
  $$u_{tt} - u_{xx} - u_{xxxx} - u_{xxxxxx} - (u^2)_{xx} - (u^2)_{xxxx} -
(uu_{xx})_{xx} - (u^3)_{xx} = 0.$$
  Such equation is analogous to original Boussinesq equation but with higher
order approximation which may ensure a more accuracy description on a long time
scale. Moreover, through a rigorous derivation from Boussiensq systems, it has
redeemed all the non-linear terms neglected in the sixth order Boussinesq
equation (SOBE),
  $$u_{tt} - u_{xx} - u_{xxxx} - u_{xxxxxx} - (u^2)_{xx} = 0.$$
  The Cauchy problem for this generalized SOBE is then considered under the
Bourgain space, $X^{s,b}$, framework. The multi-linear estimates for
$(u^2)_{xx}$, $(u^2)_{xxxx}$, $(uu_{xx})_{xx}$ and $(u^3)_{xx}$ are given, the
local wellposedness of the gSOBE is established for $s>\frac{1}{2}$.
