Jumps and cusps: a new revival effect in local dispersive PDEs

TL;DR

This paper reveals a new revival phenomenon in solutions to linear dispersive PDEs, where initial jump discontinuities reappear as both jumps and cusp singularities due to symmetry interactions.

Contribution

It introduces the concept of jump and cusp revival effects in dispersive PDEs and explicitly characterizes their formation mechanisms.

Findings

Discontinuities are revived as jump and cusp singularities.

The formation is linked to symmetries and the periodic Hilbert transform.

Explicit descriptions of the singularities are provided.

Abstract

We study the presence of a non-trivial revival effect in the solution of linear dispersive boundary value problems for two benchmark models which arise in applications: the Airy equation and the dislocated Laplacian Schr{\"o}dinger equation. In both cases, we consider boundary conditions of Dirichlet-type. We prove that, at suitable times, jump discontinuities in the initial profile are revived in the solution not only as jump discontinuities but also as logarithmic cusp singularities. We explicitly describe these singularities and show that their formation is due to interactions between the symmetries of the underlying spatial operators with the periodic Hilbert transform.