Local well-posedness for the Sixth-Order Boussinesq Equation

This work studies the local well-posedness of the initial-value problem for the nonlinear sixth-order Boussinesq equation $u_{tt}=u_{xx}+\beta u_{xxxx}+u_{xxxxxx}+(u^2)_{xx}$,where$\beta=\pm1$. We prove local well-posedness with initial data in non-homogeneous Sobolev spaces $H^s(\mathbb{R})$ for negative indices of $s\in \mathbb{R}$.