<p>The purpose of this short note is to explain how the existing results on the validity of the NLS approximation can be extended from Sobolev spaces <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42286_2025_124_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$ H^s({\mathbb {R}}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> to the spaces of functions <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42286_2025_124_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="76" /> </InlineMediaObject> <EquationSource Format="TEX">$ u = v + w $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>u</mi> <mo>=</mo> <mi>v</mi> <mo>+</mo> <mi>w</mi> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42286_2025_124_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$ v \in H_{{per}}^s $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>v</mi> <mo>∈</mo> <msubsup> <mi>H</mi> <mrow> <mrow> <mi mathvariant="italic">per</mi> </mrow> </mrow> <mi>s</mi> </msubsup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42286_2025_124_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$ w \in H^s({\mathbb {R}}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mo>∈</mo> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This allows us to use the Peregrine solution of the NLS equation to find freak or rogue wave dynamics in more complicated systems.</p>

Approximate Peregrine Solitons in Dispersive Nonlinear Wave Equations

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2025年9月1日

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Guido Schneider,
Nils Thorin

摘要

The purpose of this short note is to explain how the existing results on the validity of the NLS approximation can be extended from Sobolev spaces $ H^s({\mathbb {R}}) $ $H^{s} (R)$ to the spaces of functions $ u = v + w $ $u = v + w$ where $ v \in H_{{per}}^s $ $v \in H_{per}^{s}$ and $ w \in H^s({\mathbb {R}}) $ $w \in H^{s} (R)$ . This allows us to use the Peregrine solution of the NLS equation to find freak or rogue wave dynamics in more complicated systems.