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Jumps, cusps and fractals in the solution of the periodic linear Benjamin-Ono equation

Lyonell Boulton, Breagh Macpherson, Beatrice Pelloni

TL;DR

This work studies the periodic linear Benjamin-Ono equation on the torus and reveals a sharp regularity dichotomy: at a dense set of rational times, the solution is a finite superposition of translates of the initial data and its Hilbert transform, producing logarithmic cusps from jumps in the initial condition; for initial data of bounded variation, the solution is Hölder continuous for almost every time while its graph is fractal with upper Minkowski dimension $3/2$. A key bridge is the identity $u(x,t)=\operatorname{Re}[(I+i\mathcal{H})v(x,t)]$ relating BO to the linear Schrödinger flow $v_t=-i\partial_x^2 v$, which yields a simple, rigorous proof of cusp revivals and a Besov-space based argument for almost-everywhere regularity. The authors also derive a fractal-regularity result, showing that when $u_0\in BV$, the graph of $u(\cdot,t)$ has Minkowski dimension $3/2$ for almost all $t$, and provide numerical illustrations validating the theoretical predictions, including box-counting dimension estimates near $3/2$ for irrational times. These results deepen understanding of dispersive PDEs with jump discontinuities and connect cusp dynamics to fractal geometry via harmonic analysis.

Abstract

We establish two complementary results about the regularity of the solution of the periodic initial value problem for the linear Benjamin-Ono equation. We first give a new simple proof of the statement that, for a dense countable set of the time variable, the solution is a finite linear combination of copies of the initial condition and of its Hilbert transform. In particular, this implies that discontinuities in the initial condition are propagated in the solution as logarithmic cusps. We then show that, if the initial condition is of bounded variation (and even if it is not continuous), for almost every time the graph of the solution in space is continuous but fractal, with upper Minkowski dimension equal to 3/2. In order to illustrate this striking dichotomy, in the final section we include accurate numerical evaluations of the solution profile, as well as estimates of its box-counting dimension for two canonical choices of irrational time.

Jumps, cusps and fractals in the solution of the periodic linear Benjamin-Ono equation

TL;DR

This work studies the periodic linear Benjamin-Ono equation on the torus and reveals a sharp regularity dichotomy: at a dense set of rational times, the solution is a finite superposition of translates of the initial data and its Hilbert transform, producing logarithmic cusps from jumps in the initial condition; for initial data of bounded variation, the solution is Hölder continuous for almost every time while its graph is fractal with upper Minkowski dimension . A key bridge is the identity relating BO to the linear Schrödinger flow , which yields a simple, rigorous proof of cusp revivals and a Besov-space based argument for almost-everywhere regularity. The authors also derive a fractal-regularity result, showing that when , the graph of has Minkowski dimension for almost all , and provide numerical illustrations validating the theoretical predictions, including box-counting dimension estimates near for irrational times. These results deepen understanding of dispersive PDEs with jump discontinuities and connect cusp dynamics to fractal geometry via harmonic analysis.

Abstract

We establish two complementary results about the regularity of the solution of the periodic initial value problem for the linear Benjamin-Ono equation. We first give a new simple proof of the statement that, for a dense countable set of the time variable, the solution is a finite linear combination of copies of the initial condition and of its Hilbert transform. In particular, this implies that discontinuities in the initial condition are propagated in the solution as logarithmic cusps. We then show that, if the initial condition is of bounded variation (and even if it is not continuous), for almost every time the graph of the solution in space is continuous but fractal, with upper Minkowski dimension equal to 3/2. In order to illustrate this striking dichotomy, in the final section we include accurate numerical evaluations of the solution profile, as well as estimates of its box-counting dimension for two canonical choices of irrational time.
Paper Structure (4 sections, 4 theorems, 67 equations, 3 figures, 1 algorithm)

This paper contains 4 sections, 4 theorems, 67 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Proof of the main results
  3. Illustration of the main results
  4. The Besov spaces $\mathrm{B}^\alpha_{\infty,\infty}({\mathbb T})$

Key Result

Theorem 1

Let $u_0:\mathbb{T}\longrightarrow \mathbb{R}$. The following holds true for $u$ the solution to lBO.

Figures (3)

  • Figure 1: Solution of \ref{['lBO']} for $u_0(x)=\mathbbm{1}_{[-\frac{\pi}{2},\frac{\pi}{2}]}(x)$ at time $t=2\pi\frac{1}{3}$ superimposed on the real and imaginary parts of the solution of \ref{['lsc']}. The cusp singularities in the solution of \ref{['lBO']} correspond to jump singularities in either part of the solution of \ref{['lsc']}.
  • Figure 2: Solution for $\frac{t}{2\pi}$ a rational approximation of $\phi\sim\frac{p}{q}$ for $p=F_{16}=2584$ and $q=F_{15}=1597$. Note that $|\phi-\frac{p}{q}|<1.7\times 10^{-6}$. The estimate of the box counting dimension is $D=1.54$.
  • Figure 3: Solution for for $\frac{t}{2\pi}$ a rational approximation of $\mathrm{e}\sim\frac{p}{q}$ for $p=23225$ and $q=8544$. Note that $|\mathrm{e}-\frac{p}{q}|<6.7\times 10^{-9}$. The estimate of the box counting dimension is $D=1.46$.

Theorems & Definitions (10)

  • Theorem 1
  • Corollary 1
  • Proposition 1
  • proof
  • proof : Proof of Theorem \ref{['revfracBO']}-(a)
  • proof : Proof of Theorem \ref{['revfracBO']}-(c)
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['revfracBO']}-(b)
  • proof : Proof of Corollary \ref{['lbodim']}