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

TL;DR

This paper investigates the regularity and fractal properties of solutions to the periodic linear Benjamin-Ono equation, revealing a dichotomy between finite linear combinations at certain times and fractal, continuous graphs at others.

Contribution

It provides a new proof of solution structure at specific times and demonstrates fractal behavior of solutions for almost every time, highlighting complex solution regularity.

Findings

Solutions are finite linear combinations of initial data at dense countable times.

Discontinuities propagate as logarithmic cusps in the solution.

Solutions are fractal with Minkowski dimension 3/2 at almost every time.

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.