Properties of solutions to the Camassa-Holm equation on the line in a class containing the peakons

TL;DR

This paper investigates the regularity and decay properties of solutions to the Camassa-Holm equation on the line, especially those including peakon solutions, showing how initial regularity is preserved and decay results extend.

Contribution

It demonstrates the transfer of initial regularity to solutions with peakons and extends decay results to a broader class of solutions.

Findings

Regularity of initial data is preserved in solutions containing peakons.

Decay properties from previous studies are extended to the class of solutions considered.

Solution regularity resembles that of inviscid Burgers' equation with similar initial data.

Abstract

We study special properties of solutions to the IVP associated to the Camassa-Holm equation on the line related to the regularity and the decay of solutions. The first aim is to show how the regularity on the initial data is transferred to the corresponding solution in a class containing the "peakon solutions". In particular, we shall show that the local regularity is similar to that exhibited by the solution of the inviscid Burger's equation with the same initial datum. The second goal is to prove that the decay results obtained in a paper of Himonas, Misio{\l}ek, Ponce, and Zhou extend to the class of solutions considered here.