<p>In this paper, we prove an abstract Birkhoff normal form theorem for some unbounded infinite dimensional Hamiltonian systems. Based on this result we obtain that the solution to Derivative Nonlinear Schrödinger equations under periodic boundary condition with typical small enough initial value remains small in the Sobolev norm <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2109_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$ H^{\textbf{s}}(\mathbb {T})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi mathvariant="bold">s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> over a long time interval. The length of the time interval is equal to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2109_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$e^{|\ln R|^{1+\gamma }}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <msup> <mrow> <mo stretchy="false">|</mo> <mo>ln</mo> <mi>R</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>γ</mi> </mrow> </msup> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2109_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="89" /> </InlineMediaObject> <EquationSource Format="TEX">$0<\gamma <1/5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>γ</mi> <mo><</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation> as the initial value is smaller than <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="205_2025_2109_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$R\ll 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo>≪</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

Long Time Stability of Hamiltonian Derivative Nonlinear Schrödinger Equations Without Potential

期刊论文
发表日期：

2025年6月6日

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Hu Shengqing,
Zhang Jing

摘要

In this paper, we prove an abstract Birkhoff normal form theorem for some unbounded infinite dimensional Hamiltonian systems. Based on this result we obtain that the solution to Derivative Nonlinear Schrödinger equations under periodic boundary condition with typical small enough initial value remains small in the Sobolev norm $ H^{\textbf{s}}(\mathbb {T})$ $H^{s} (T)$ over a long time interval. The length of the time interval is equal to $e^{|\ln R|^{1+\gamma }}$ $e^{{| ln R |}^{1 + γ}}$ with $0<\gamma <1/5$ $0 < γ < 1 / 5$ as the initial value is smaller than $R\ll 1$ $R ≪ 1$ .