Local well-posedness and decay for some generalized shallow water equations

Consideration herein is a big class of nonlocal partial differential equations including some classical shallow water models such as b-family equation, Novikov equation, rotation-Camassa-Holm equation, generalized Camassa-Holm equation and Camassa-Holm-Novikov equation over a whole line. Based on the local structure of the dynamics along the characteristics, a local well-posedness result is established for some initial data class containing certain non-smooth solitary wave solutions. Subsequently in this set certain decay persistence properties are also constructed. These partially extend some obtained results and bring the effect of nonlinearities to more extent.