Dispersive Revival Phenomena for Two-Dimensional Dispersive Evolution Equations

Abstract

In this paper, we investigate dispersive revival phenomena of two-dimensional linear spatially periodic dispersive evolution equations, defined on a rectangle with periodic boundary conditions and discontinuous initial profiles. We begin by studying the periodic initial-boundary value problem for general two-dimensional dispersive evolution equations. We prove that, when posed on a periodic rational torus, two-dimensional linear dispersive equations with homogeneous power integral binary polynomial dispersion relations exhibit the standard dispersive revival effect at rational times. This means that the resulting solution can be expressed as a finite linear combination of translates of the initial data. Next, we explore a novel revival phenomenon in two-dimensional equations with nonpolynomial dispersion relations, in the concrete case of the periodic initial-boundary value problem for the linear Kadomtsev–Petviashvili equation on a square with step function initial data. In this scenario, the revival phenomenon exhibits a novel characteristic that there are radically different qualitative behaviors in the $x$- and $y$-directions. We provide an analytic description of this dichotomous revival phenomenon and present illustrative numerical simulations.