<p>We study the global in time existence of small solutions to the Cauchy problem for the fractional nonlinear Schrödinger equation of order <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_344_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="73" /> </InlineMediaObject> <EquationSource Format="TEX">$ \alpha \in \left( \frac{3}{2},3\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mfenced close=")" open="("> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> <mo>,</mo> <mn>3</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>. We consider the cubic derivative nonlinearity with a time growth of order <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_344_Article_IEq2.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="85" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu \in \left( 0,\frac{1}{24} \right) .$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>∈</mo> <mfenced close=")" open="("> <mn>0</mn> <mo>,</mo> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mfenced> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We remark that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="42985_2025_344_Article_IEq3.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$\nu >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> means that the problem is considered as subcritical case in the sense of the large time asymptotic behavior of solutions. We assume that the initial data have an analytic extension on the sector and are small, then we find the large time asymptotics of the solutions with a phase correction.</p>

Large time asymptotics for the fractional Schrödinger equation with subcritical derivative nonlinearities

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发表日期：

2025年8月19日

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Nakao Hayashi,
Pavel I. Naumkin

摘要

We study the global in time existence of small solutions to the Cauchy problem for the fractional nonlinear Schrödinger equation of order $ \alpha \in \left( \frac{3}{2},3\right) $ $α \in (\frac{3}{2}, 3)$ . We consider the cubic derivative nonlinearity with a time growth of order $\nu \in \left( 0,\frac{1}{24} \right) .$ $ν \in (0, \frac{1}{24}) .$ We remark that $\nu >0$ $ν > 0$ means that the problem is considered as subcritical case in the sense of the large time asymptotic behavior of solutions. We assume that the initial data have an analytic extension on the sector and are small, then we find the large time asymptotics of the solutions with a phase correction.