New Revival Phenomena for Linear Integro-Differential Equations

TL;DR

This paper explores a new revival phenomenon in linear periodic integro-differential equations, showing how solutions form dispersively quantised cusps at rational times, supported by analysis and simulations.

Contribution

It introduces a novel revival phenomenon in nonlocal linear equations with convolution kernels, expanding understanding of wave behavior in water wave models.

Findings

Revival manifests as dispersively quantised cusped solutions at rational times.

Analytic description of the revival phenomenon is provided.

Numerical simulations illustrate the theoretical findings.

Abstract

We present and analyse a novel manifestation of the revival phenomenon for linear spatially periodic evolution equations, in the concrete case of three nonlocal equations that arise in water wave theory and are defined by convolution kernels. Revival in these cases is manifested in the form of dispersively quantised cusped solutions at rational times. We give an analytic description of this phenomenon, and present illustrative numerical simulations.