The Cauchy problem for the integrable RZQ equation

TL;DR

This paper investigates the well-posedness of the initial value problem for the integrable RZQ equation, a fifth-order Camassa-Holm type equation, establishing conditions for existence, uniqueness, and continuity of solutions in Sobolev spaces.

Contribution

It proves the well-posedness of the RZQ equation in Sobolev spaces for s>7/2 and demonstrates ill-posedness for s<7/2, providing new insights into its mathematical properties.

Findings

Well-posed in Sobolev spaces $H^s$ for $s>7/2$

Data-to-solution map is not uniformly continuous in $H^s$

Ill-posed in $H^s$ for $s<7/2$

Abstract

In this paper we study a new integrable fifth-order Camassa-Holm (CH)-type equation derived by Reyes, Zhu, and Qiao, which we call the RZQ equation. The m-form of this equation possesses a striking similarity to the m-form of the CH equation. However, unlike the CH equation, the nonlocal form of this equation cannot be interpreted as a nonlocal perturbation of Burgers' equation. We prove that the initial value problem corresponding to the RZQ equation is well-posed in the sense of Hadamard, in Sobolev spaces $H^s$, $s>7/2$. We further show that the data-to-solution map is not uniformly continuous in the $H^s$ topology, though it is H\"older continuous in a weaker topology. The initial value problem corresponding to the RZQ equation is ill-posed in $H^s$ for $s<7/2$.