On the two-component b-family system modeling shallow-water waves of medium amplitude

We formally derive a two-component b-family system from the full governing equations for two-dimensional water waves in the medium-amplitude regime. The system contains higher-order nonlinearities and includes a two-component Camassa-Holm type system for equatorial shallow-water waves with the Coriolis effect as a special case. We establish the local well-posedness of the Cauchy problem in Bˢ⁻¹p,r× Bˢp,r with s>max\1+1p,32\. Using Littlewood-Paley theory and Osgood's lemma, we also prove local well-posedness in the critical Besov space B¹²2,1× B³²2,1. Furthermore, we derive a blow-up criterion for strong solutions. For a special choice of the parameters, we prove finite-time blow-up for a class of initial data and provide an upper bound for the maximal existence time.