<p>The modified Camassa–Holm equation with cubic nonlinearity is a completely integrable model, utilized to describe the unidirectional propagation of shallow-water waves. In this study, we show that the 2-soliton, when employed as a solution to the initial-value problem for the modified Camassa–Holm equation, is nonlinearly stable to perturbations within the Sobolev space <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3113_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$H^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> related to the momentum density. This stability is demonstrated through the conservation of crucial quantities in the Sobolev spaces <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3113_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$H^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3113_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$H^{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3113_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation> in terms of the momentum density.</p>

Stability of 2-soliton solutions in the modified Camassa–Holm equation

期刊论文
发表日期：

2025年2月24日

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Ji Li,
Yue Liu,
Guangming Zhu

摘要

The modified Camassa–Holm equation with cubic nonlinearity is a completely integrable model, utilized to describe the unidirectional propagation of shallow-water waves. In this study, we show that the 2-soliton, when employed as a solution to the initial-value problem for the modified Camassa–Holm equation, is nonlinearly stable to perturbations within the Sobolev space $H^2$ $H^{2}$ related to the momentum density. This stability is demonstrated through the conservation of crucial quantities in the Sobolev spaces $H^2$ $H^{2}$ , $H^{1}$ $H^{1}$ , and $L^{1}$ $L^{1}$ in terms of the momentum density.