On the solutions of some nonlocal models for nonlinear dispersive waves

Abstract

In this paper we study special properties of solutions of the initial value problem associated to a class of nonlinear dispersive equations where the operator modelling dispersive effects is nonlocal. In particular, we prove that solutions of the surface tension Whitham equations posed on the real line satisfy the propagation of regularity phenomena, which says that regularity of the initial data on the right hand side of the real line is propagated to the left hand side by the flow solution. A similar result is obtained for solutions of the Full Dispersion Kadomtsev–Petviashvili equation, a natural (weakly transverse) two-dimensional version of the Whitham equation, with and without surface tension. We establish that the augmented regularity of the initial data on certain distinguished subsets of the Euclidean space is transmitted by the flow solution at an infinite rate. The underlying approach involves treating the general equation as a perturbed version of a class of fractional equations with well-established properties.