An integrable pseudospherical equation with pseudo-peakon solutions

TL;DR

This paper introduces a new integrable equation related to pseudospherical surfaces, develops methods for constructing explicit solutions including pseudo-peakons, and connects it to the Degasperis-Procesi equation via a Miura-type transformation.

Contribution

It presents a novel integrable pseudospherical equation with explicit pseudo-peakon solutions and introduces the collage method for constructing weak solutions, also establishing a link to the Degasperis-Procesi equation.

Findings

Existence of explicit pseudo-peakon solutions.

Development of the collage method for smooth joining of solutions.

Connection to the Degasperis-Procesi equation via Miura transformation.

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.