On the Cauchy Problem for the New Shallow-water Models with Cubic Nonlinearity

This paper is devoted to the new shallow-water model with cubic nonlinearity, which admits the single peaked solitons and multi-peakon solutions, and includes both the modified Camassa–Holm equation (also called Fokas–Olver–Rosenau–Qiao equation) and the Novikov equation as two special cases. On the one hand, based on a generalized Ovsyannikov type theorem, we prove the existence and uniqueness of solutions in the Gevrey–Sobolev spaces with the lower bound of the lifespan, and show the continuity of the data-to-solution map for the system. On the other hand, we prove the persistence properties in weighted spaces of the solution, provided that the initial potential satisfies a certain sign condition.