<p>We investigate the global in time existence of small solutions to the Cauchy problem for the subcritical fractional derivative nonlinear Schrödinger equation <Equation ID="Equ20"> <EquationSource Format="TEX">$ \left\{ \begin{array}{c} i\partial _{t}u-\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha } u=it^{\nu }\partial _{x}\left( \left| u\right| ^{2}u\right) ,\text t>0\textbf{,}x\in \mathbb {R},\\ u\left( 0,x\right) =u_{0}\left( x\right) ,\,x\in \mathbb {R}\textbf{,} \end{array} \right. $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd> <mrow> <mi>i</mi> <msub> <mi>∂</mi> <mi>t</mi> </msub> <mi>u</mi> <mo>-</mo> <mfrac> <mn>1</mn> <mi>α</mi> </mfrac> <msup> <mfenced close="|" open="|"> <msub> <mi>∂</mi> <mi>x</mi> </msub> </mfenced> <mi>α</mi> </msup> <mi>u</mi> <mo>=</mo> <mi>i</mi> <msup> <mi>t</mi> <mi>ν</mi> </msup> <msub> <mi>∂</mi> <mi>x</mi> </msub> <mfenced close=")" open="("> <msup> <mfenced close="|" open="|"> <mi>u</mi> </mfenced> <mn>2</mn> </msup> <mi>u</mi> </mfenced> <mo>,</mo> <mtext>t</mtext> <mo>></mo> <mn>0</mn> <mo>,</mo> <mi>x</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow /> <mi>u</mi> <mfenced close=")" open="("> <mn>0</mn> <mo>,</mo> <mi>x</mi> </mfenced> <mo>=</mo> <msub> <mi>u</mi> <mn>0</mn> </msub> <mfenced close=")" open="("> <mi>x</mi> </mfenced> <mo>,</mo> <mspace width="0.166667em" /> <mi>x</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </math></EquationSource> </Equation>where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha \in \left( 0,1\right) \cup \left( 1,\frac{3}{2}\right] $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mfenced close=")" open="("> <mn>0</mn> <mo>,</mo> <mn>1</mn> </mfenced> <mo>∪</mo> <mfenced close="]" open="("> <mn>1</mn> <mo>,</mo> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mfenced> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$0<\nu <\frac{1}{24}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>ν</mi> <mo><</mo> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>. The parameter <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\nu >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> reflects the subcritical nature of the nonlinearity with respect to the large time asymptotic behavior of solutions. We assume that the initial data <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$u_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>u</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> admits a small analytic extension in a suitable sector of the complex plane. Under these assumptions, we derive the precise large time asymptotics of the global solution.</p>

Modified scattering for subcritical fractional dNLS equations with order $\alpha \in \left( 0,1\right) \cup \left( 1,\frac{3}{2}\right] $

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2026年1月7日

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Nakao Hayashi,
Pavel I. Naumkin

摘要

We investigate the global in time existence of small solutions to the Cauchy problem for the subcritical fractional derivative nonlinear Schrödinger equation $ \left\{ \begin{array}{c} i\partial _{t}u-\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha } u=it^{\nu }\partial _{x}\left( \left| u\right| ^{2}u\right) ,\text t>0\textbf{,}x\in \mathbb {R},\\ u\left( 0,x\right) =u_{0}\left( x\right) ,\,x\in \mathbb {R}\textbf{,} \end{array} \right. $ $\{\begin{matrix} i \partial_{t} u - \frac{1}{α} {|\partial_{x}|}^{α} u = i t^{ν} \partial_{x} ({|u|}^{2}, u), t > 0, x \in R, \\ u (0, x) = u_{0} (x), x \in R, \end{matrix})$ where $\alpha \in \left( 0,1\right) \cup \left( 1,\frac{3}{2}\right] $ $α \in (0, 1) \cup (1, \frac{3}{2}]$ and $0<\nu <\frac{1}{24}$ $0 < ν < \frac{1}{24}$ . The parameter $\nu >0$ $ν > 0$ reflects the subcritical nature of the nonlinearity with respect to the large time asymptotic behavior of solutions. We assume that the initial data $u_{0}$ $u_{0}$ admits a small analytic extension in a suitable sector of the complex plane. Under these assumptions, we derive the precise large time asymptotics of the global solution.

Modified scattering for subcritical fractional dNLS equations with order \(\alpha \in \left( 0,1\right) \cup \left( 1,\frac{3}{2}\right] \)

摘要