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<p>Spectral stability analysis of ”anomalous” solitons and multi-solitons is presented in the context of a generalized Hamiltonian system called the Kaup-Kupershmidt (KK) equation. The KK equation is a completely integrable fifth order Korteweg-de Vries equation, which admits third order eigenvalue problem in its Lax pair. We also prove Hamiltonian-Krein index identities in verifying stability criterion of its multi-solitons. However, the KK equation does not possess the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_942_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> conservation law and the linearized operators around the multi-solitons have no spectral gap. The main ingredients of the proof are new operator identities for second variation operator and completeness in <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="21_2025_942_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> of the squared eigenfunctions of the third order eigenvalue problem for the KK equation. The operator identities and completeness relation are shown by employing the recursion operators of the KK equation.</p>

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Spectral Stability of Multi-Solitons for the Kaup-Kupershmidt Equation

  • 期刊论文

  • 发表日期：

    2025年5月14日

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  • Zhong Wang

摘要

Spectral stability analysis of ”anomalous” solitons and multi-solitons is presented in the context of a generalized Hamiltonian system called the Kaup-Kupershmidt (KK) equation. The KK equation is a completely integrable fifth order Korteweg-de Vries equation, which admits third order eigenvalue problem in its Lax pair. We also prove Hamiltonian-Krein index identities in verifying stability criterion of its multi-solitons. However, the KK equation does not possess the \(L^2\) L 2 conservation law and the linearized operators around the multi-solitons have no spectral gap. The main ingredients of the proof are new operator identities for second variation operator and completeness in \(L^2\) L 2 of the squared eigenfunctions of the third order eigenvalue problem for the KK equation. The operator identities and completeness relation are shown by employing the recursion operators of the KK equation.