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<p>In this paper, we establish the existence of the global weak admissible solution for the Cauchy problem of a <i>N</i>-peakon system in the sense of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_963_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^1(\mathbb {R})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> space under a sign condition. Second, we claim that the global weak admissible solution for the system with the same initial data is not unique by giving a example. Finally, an image of the solutions of the above example which does not satisfy the uniqueness is given, which makes it easier to see the properties of non-uniqueness more intuitively.</p>

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The Existence and Non-Uniqueness of Global Weak Solution to a New Integrable System in \(H^1(\mathbb {R})\)

  • 期刊论文

  • 发表日期：

    2025年7月18日

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  • Pei Zheng,
  • Zhaoyang Yin

摘要

In this paper, we establish the existence of the global weak admissible solution for the Cauchy problem of a N-peakon system in the sense of \(H^1(\mathbb {R})\) H 1 ( R ) space under a sign condition. Second, we claim that the global weak admissible solution for the system with the same initial data is not unique by giving a example. Finally, an image of the solutions of the above example which does not satisfy the uniqueness is given, which makes it easier to see the properties of non-uniqueness more intuitively.