Pseudo-peakons and Cauchy analysis for an integrable fifth-order equation of Camassa-Holm type

TL;DR

This paper explores a fifth-order Camassa-Holm type equation, demonstrating its integrability, pseudo-peakon solutions, and analyzing the well-posedness and blow-up conditions of its Cauchy problem.

Contribution

It introduces a new fifth-order CH-type equation with pseudo-peakon solutions and provides a comprehensive analysis of its integrability, solutions, and well-posedness.

Findings

Existence of pseudo-peakon solutions for the fifth-order equation

Local well-posedness in Sobolev spaces for initial data in H^s, s > 7/2

Conditions for global existence and finite-time blow-up

Abstract

In this paper we discuss integrable higher order equations {\em of Camassa-Holm (CH) type}. Our higher order CH-type equations are "geometrically integrable", that is, they describe one-parametric families of pseudo-spherical surfaces, in a sense explained in Section 1, and they are integrable in the sense of zero curvature formulation ($\simeq$ Lax pair) with infinitely many local conservation laws. The major focus of the present paper is on a specific fifth order CH-type equation admitting {\em pseudo-peakons} solutions, that is, weak bounded solutions with differentiable first derivative and continuous and bounded second derivative, but such that any higher order derivative blows up. Furthermore, we investigate the Cauchy problem of this fifth order CH-type equation on the real line and prove local well-posedness under the initial conditions $u_0 \in H^s(\mathbb{R})$, $s > 7/2$. In addition, we study conditions for global well-posedness in $H^4(\mathbb{R})$ as well as conditions causing local solutions to blow up in a finite time. We conclude our paper with some comments on the geometric content of the high order CH-type equations.