On the initial value problem for a class of nonlinear dispersive models (Q1323169)

The initial value problem is investigated for a class of equations \[ u_{tt}=- u_{xxx}+ (\psi(u,u_ x))_ x+ C(u),\;t\in [0,T],\quad u(x,0)= f(x),\quad u_ t(x,0)= g(x), \tag{1} \] where \(\psi\), \(C\) are real functions. The result concerning local well-posedness is established. When \((\psi(u, u_ x))_ x= [u_ x^ 2 ]_ x\), \(C(u)=0\), equation (1) coincides with the Zabusky equation. When \((\psi(u, u_ x))_ x =0\), \(C(u)= u^ 3-u\) equation (1) coincides with the Bretherton equation.