New Revival Phenomena for Bidirectional Dispersive Hyperbolic Equations

TL;DR

This paper investigates revival and fractalization phenomena in bidirectional dispersive hyperbolic equations on bounded intervals, revealing how initial data and dispersion relations influence complex solution behaviors, including in nonlinear cases.

Contribution

It extends the analysis of dispersive revival and fractalization phenomena to general bidirectional dispersive equations with bounded, discontinuous initial data, linking asymptotics to observed phenomena.

Findings

Dispersive revival occurs at rational times for step initial data.

Non-polynomial dispersion relations lead to fractal profiles at all times.

Numerical experiments confirm phenomena in nonlinear beam equations.

Abstract

In this paper, the dispersive revival and fractalization phenomena for bidirectional dispersive equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles are investigated. Firstly, we study the periodic initial-boundary value problem of the linear beam equation with step function initial data, and analyze the manifestation of the revival phenomenon for the corresponding solution at rational times. Next, we extend the investigation to periodic initial-boundary value problems of more general bidirectional dispersive equations. We prove that, if the initial functions are of bounded variation, the dynamical evolution of such periodic problems depend essentially upon the large wave number asymptotics of the associated dispersion relations. Integral polynomial or asymptotically integral polynomial dispersion relations produce dispersive revival/fractalization rational/irrational dichotomies, whereas those with non-polynomial growth result in fractal profiles at all times. Finally, numerical experiments, in the concrete case of the nonlinear beam equation, are used to demonstrate how such effects persist into the nonlinear regime.