A new highly nonlinear equation modelling shallow-water waves with constant vorticity

In this paper we apply the approach of formal asymptotic expansions and perturbation theory to derive a new highly nonlinear shallow-water model from the full governing equations for two dimensional incompressible fluid with constant vorticity. This approximate model is generated by introduction of a larger scaling than the Camassa-Holm one, which is shown to be optimal in the sense that there are no non-local terms appearing in free surface equation. Moreover, we establish the local well-posedness of the Cauchy problem in Besov spaces, and give a blow-up criterion, which improve the previous corresponding results in Sobolev spaces.