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, uₜₜ - uₓₓ - uₓₓₓₓ - uₓₓₓₓₓₓ - (u²)ₓₓ - (u²)ₓₓₓₓ - (uuₓₓ)ₓₓ - (u³)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), uₜₜ - uₓₓ - uₓₓₓₓ - uₓₓₓₓₓₓ - (u²)ₓₓ = 0. The Cauchy problem for this generalized SOBE is then considered under the Bourgain space, Xs,b, framework. The multi-linear estimates for (u²)ₓₓ, (u²)ₓₓₓₓ, (uuₓₓ)ₓₓ and (u³)ₓₓ are given, the local wellposedness of the gSOBE is established for s>1/2.