<p>In this paper, we study a new integrable fifth-order Camassa–Holm (CH)-type equation derived by Reyes, Zhu, and Qiao [<CitationRef CitationID="CR49">49</CitationRef>], 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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H^s$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>s</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$s>7/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>></mo> <mn>7</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. We further show that the data-to-solution map is not uniformly continuous in the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^s$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>s</mi> </msup> </math></EquationSource> </InlineEquation> topology, though it is Hölder continuous in a weaker topology. The initial value problem corresponding to the RZQ equation is ill-posed in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H^s$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mi>s</mi> </msup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$s<7/2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo><</mo> <mn>7</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

The Cauchy problem for the integrable RZQ equation

期刊论文
发表日期：

2026年7月9日

查看全文

John Holmes,
Kathryn Massey,
Ryan C. Thompson

摘要

In this paper, we study a new integrable fifth-order Camassa–Holm (CH)-type equation derived by Reyes, Zhu, and Qiao [49], 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 $H^s$ $H^{s}$ , $s>7/2$ $s > 7 / 2$ . We further show that the data-to-solution map is not uniformly continuous in the $H^s$ $H^{s}$ topology, though it is Hölder continuous in a weaker topology. The initial value problem corresponding to the RZQ equation is ill-posed in $H^s$ $H^{s}$ for $s<7/2$ $s < 7 / 2$ .