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The Cauchy problem for the integrable RZQ equation

John Holmes, Katie Massey, Ryan C. Thompson

TL;DR

The paper analyzes the integrable RZQ equation, a fifth-order Camassa-Holm–type model, establishing Hadamard well-posedness in Sobolev spaces $H^s$ for $s>7/2$ via a Galerkin/mollification framework and proving ill-posedness for $s<7/2$ on the real line. It demonstrates that the data-to-solution map is not uniformly continuous but is Hölder continuous in weaker Sobolev topologies, by constructing approximate solutions and comparing actual versus approximate dynamics on both $\mathbb{T}$ and $\mathbb{R}$. The work provides a sharp regularity threshold at $s=7/2$, develops a robust mollification-based existence/uniqueness theory, and extends to detailed nonuniform dependence results for both periodic and non-periodic settings, highlighting the role of nonlocal structure in higher-order CH-type equations.

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 Hölder continuous in a weaker topology. The initial value problem corresponding to the RZQ equation is ill-posed in $H^s$ for $s<7/2$.

The Cauchy problem for the integrable RZQ equation

TL;DR

The paper analyzes the integrable RZQ equation, a fifth-order Camassa-Holm–type model, establishing Hadamard well-posedness in Sobolev spaces for via a Galerkin/mollification framework and proving ill-posedness for on the real line. It demonstrates that the data-to-solution map is not uniformly continuous but is Hölder continuous in weaker Sobolev topologies, by constructing approximate solutions and comparing actual versus approximate dynamics on both and . The work provides a sharp regularity threshold at , develops a robust mollification-based existence/uniqueness theory, and extends to detailed nonuniform dependence results for both periodic and non-periodic settings, highlighting the role of nonlocal structure in higher-order CH-type equations.

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 , . We further show that the data-to-solution map is not uniformly continuous in the topology, though it is Hölder continuous in a weaker topology. The initial value problem corresponding to the RZQ equation is ill-posed in for .

Paper Structure

This paper contains 19 sections, 21 theorems, 220 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Proof of Well-posedness
  3. Existence of Solutions
  4. Weak$^*$ convergence in $L^\infty([0,T]; H^s)$
  5. Strong convergence in $C([0,T]; H^{s-1})$
  6. Strong convergence in $C([0,T]; H^{s-\sigma})$
  7. Convergence in $C^3$
  8. Verifying that the limit $u$ solves the RZQ equation
  9. Improving the regularity to $C([0,T];H^s)$
  10. Uniqueness of Solutions
  11. Continuous Dependence
  12. Continuity of the data-to-solution map
  13. Nonuniform dependence on $\mathbb T$
  14. Nonuniform dependence on $\mathbb R$
  15. Estimating the error.
  16. ...and 4 more sections

Key Result

Theorem 1.1

In both the periodic and non-periodic cases, the initial value problem for the RZQ equation is locally well-posed in $H^s$ for $s>7/2$. Furthermore, there exists a $T>0$ which depends only upon the size of the initial data, $\|u_0\|_{H^s}$, and $s$, such that for all $0\le t \le T$, the solution sat

Figures (3)

  • Figure 1: Left: CH peakon \ref{['ch-peakon']}; right: is the RZQ pseudo-peakon \ref{['qr-peakon']}.
  • Figure 2: The first (left) and second (right) derivatives of the RZQ pseudo-peakon \ref{['qr-peakon']}.
  • Figure 3: Regions $A_1$, $A_2$ and $A_3$.

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 1.1
  • Theorem 2.1
  • proof
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • ...and 18 more