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<p>We prove the asymptotic stability of the high speed solitary waves to the Benjamin equation. This is done by establishing a Liouville property for the nonlinear evolution of the Benjamin equation around these solitary waves. To do this, inspired by [<CitationRef CitationID="CR21">21</CitationRef>], we make use of the KdV limit of the Benjamin equation together with known rigidity property of the KdV flow. The main difficulties are linked to the presence of the non-local operator <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10884_2025_10461_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\( {{\mathcal {H}}}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> as well as the non-positivity of the quadratic part of the energy in the case <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10884_2025_10461_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\( \gamma &lt;0\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> which corresponds to the physical case.</p>

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Asymptotic Stability of Fast Solitary Waves to the Benjamin Equation

  • 期刊论文

  • 发表日期：

    2025年9月12日

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  • May Abdallah,
  • Mohamad Darwich,
  • Luc Molinet

摘要

We prove the asymptotic stability of the high speed solitary waves to the Benjamin equation. This is done by establishing a Liouville property for the nonlinear evolution of the Benjamin equation around these solitary waves. To do this, inspired by [21], we make use of the KdV limit of the Benjamin equation together with known rigidity property of the KdV flow. The main difficulties are linked to the presence of the non-local operator \( {{\mathcal {H}}}\) H as well as the non-positivity of the quadratic part of the energy in the case \( \gamma <0\) γ < 0 which corresponds to the physical case.