Dispersive Lamb Systems

TL;DR

This paper investigates how dispersive Lamb systems respond under periodic conditions, revealing that sublinear dispersion relations lead to fractal solutions, contrasting with the superlinear case associated with dispersive quantization effects.

Contribution

It uncovers the contrasting behaviors of dispersive Lamb systems under different asymptotic dispersion regimes, highlighting the emergence of fractal solutions in the sublinear case.

Findings

Fractal solution profiles emerge in sublinear dispersion regimes.

Linear or superlinear dispersion relations produce smooth responses.

Contrasts with the Talbot effect in unforced media with discontinuous initial conditions.

Abstract

Under periodic boundary conditions, a one-dimensional dispersive medium driven by a Lamb oscillator exhibits a smooth response when the dispersion relation is asymptotically linear or superlinear at large wave numbers, but unusual fractal solution profile emerge when the dispersion relation is asymptotically sublinear. Strikingly, this is precisely the opposite to the superlinear asymptotic regime required for fractalization and dispersive quantization, also known as the Talbot effect, of the unforced medium produced by discontinuous initial conditions.