New Revival Phenomena for Bidirectional Dispersive Hyperbolic Equations

TL;DR

This work identifies a new revival/fractalization dichotomy for bidirectional dispersive equations on periodic domains, where initial data with bounded variation yield dispersive quantization at rational times and fractal profiles at irrational times, generalizing prior unidirectional results. It provides explicit representations for the linear beam equation and extends the analysis to general bidirectional dispersive laws, including monomial, integral-polynomial, and non-polynomial dispersions, showing that revivals arise as sums of translates of initial data plus regular corrective terms. The authors further demonstrate that these phenomena persist into nonlinear regimes through Fourier-spectral-method simulations of the nonlinear beam equation, with nonlinearity affecting the convexity and finer shape of revival profiles while leaving the qualitative revival structure intact. The work also uncovers connections to number-theoretic objects such as the Riemann zeta function at specific rational times and highlights practical implications for understanding wave revival in bounded, periodic settings with rough initial data.

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.

Figures (11)

• Figure 1: The solutions to the periodic initial-boundary value problem for the linear beam equation at rational times.
• Figure 2: The solutions to the periodic initial-boundary value problem for the linear beam equation at irrational times.
• Figure 3: The solutions to the periodic initial-boundary value problem \ref{['ibv-beam']} with initial data $f(x)=\tilde{\sigma}(x),\,g(x)=\sigma(x)$.
• Figure 4: The solutions to the periodic initial-boundary value problem \ref{['ibv-beam']} with initial data $f(x)=0,\,g(x)=\sigma(x)$.
• Figure 5: The solutions to the periodic initial-boundary value problem \ref{['ibv-beam']} with initial data $f(x)=\sigma(x),\,g(x)=0$.

Theorems & Definitions (21)

• Lemma 2.1
• Lemma 2.2
• Definition 2.3
• Theorem 2.4
• Remark 2.5
• proof : Proof of Lemma \ref{['beam-le1']}
• Theorem 2.6
• Remark 2.7
• Remark 2.8
• Theorem 3.1