An integrable pseudospherical equation with pseudo-peakon solutions
Abstract
We study an integrable equation whose solutions define a triad of one-forms describing a surface with Gaussian curvature -1. We identify a local group of diffeomorphisms that preserve these solutions and establish conserved quantities. From the symmetries, we obtain invariant solutions that provide explicit metrics for the surfaces. These solutions are unbounded and often appear in mirrored pairs. We introduce the ``collage'' method, which uses conserved quantities to remove unbounded parts and smoothly join the solutions, leading to weak solutions consistent with the conserved quantities. As a result we get pseudo-peakons, which are smoother than Camassa-Holm peakons. Additionally, we apply a Miura-type transformation to relate our equation to the Degasperis-Procesi equation, allowing us to recover peakon and shock-peakon solutions for it from the solutions of the other equation.
Cite
@article{arxiv.2409.01537,
title = {An integrable pseudospherical equation with pseudo-peakon solutions},
author = {Priscila Leal da Silva and Igor Leite Freire and Nazime Sales Filho},
journal= {arXiv preprint arXiv:2409.01537},
year = {2024}
}
Comments
Parts of the original version concerning shock-peakons have been update, whereas references [9] and [10] have been added as well. We thank Professor Hans Lundmark for his valuable comments and suggestions made on the earlier version of the manuscript