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

Authors

Enrique G. Reyes
Mingxuan Zhu
Zhijun Qiao, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

11-15-2021

Abstract

In this paper we introduce a hierarchy of integrable higher order equations of Camassa-Holm (CH) type, that is, we present infinitely many nonlinear equations depending on inertia operators which generalize the standard momentum operator A2=∂xx−1 appearing in the Camassa-Holm equation mt=−mxu−2mux, m=A2(u). Our higher order CH-type equations are integrable in the sense that they possess an infinite number of local conservation laws, quadratic pseudo-potentials, and zero curvature formulations. We focus mainly on the fifth order CH-type equation and we show that it admits {\em pseudo-peakons}, this is, bounded solutions with differentiable first derivative and continuous and bounded second derivative, but whose higher order derivatives blow up. Furthermore, we investigate the Cauchy problem of our fifth order equation on the real line and prove local well-posedness for initial conditions u0∈Hs(R), s>7/2. In addition, we discuss conditions for global well-posedness in H4(R) as well as conditions causing local solutions to blow up in a finite time. We finish our paper with some comments on the geometric content of our equations of CH-type.

Recommended Citation

Reyes, Enrique G., Mingxuan Zhu, and Zhijun Qiao. "Pseudo-peakons and Cauchy analysis for an integrable fifth-order equation of Camassa-Holm type." arXiv preprint arXiv:2111.07733 (2021).