Local well-posedness for a generalized sixth-order Boussinesq equation

A formally second order correct Boussinesq-type equation that describes unidirectional shallow water waves is derived, $$u_{tt}-u_{xx}-u_{xxxx}-u_{xxxxxx}-(u^2{)}_{xx}-(u^2{)}_{xxxx}-(u{u}_{xx}{)}_{xx}-(u^3{)}_{xx}=0.$$ 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 $(u^2{)}_{xx}$, $(u^2{)}_{xxxx}$, $(u{u}_{xx}{)}_{xx}$ and $(u^3{)}_{xx}$ are given, the local wellposedness of the gSOBE is established for $s>\frac{1}{2}$.