Journal of Evolution Equations ( IF 1.4 ) Pub Date : 2023-08-21 , DOI: 10.1007/s00028-023-00910-1 Nakao Hayashi , Pavel I. Naumkin
We study the large-time asymptotics of solutions to the fractional modified Korteweg–de Vries equation
$$\begin{aligned} \left\{ \begin{array}{c} \partial _{t}u+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha -1}\partial _{x}u=\partial _{x}\left( u^{3}\right) ,~ t>0,\ x\in {\mathbb {R}}\textbf{,} \\ u\left( 0,x\right) =u_{0}\left( x\right) ,\ x\in {\mathbb {R}}\textbf{,} \end{array} \right. \end{aligned}$$(0.1)where \(\alpha \in \left( 1,2\right) ,\) \(\left| \partial _{x}\right| ^{\alpha }={\mathcal {F}}^{-1}\left| \xi \right| ^{\alpha }{\mathcal {F}}\) is the fractional derivative. The case of \(\alpha =3\) corresponds to the classical modified KdV equation. In the case of \(\alpha =2\), it is the modified Benjamin–Ono equation. Our aim is to extend the results in [10, 16] for \(\alpha \in \left( 0,1\right) \) to \(\alpha \in \left( 1,2\right) \). We develop the method based on the factorization techniques, which was started in [11], and apply the known results on the \({\textbf{L}}^{2}\) - boundedness of pseudodifferential operators to get the large-time asymptotics of solutions.
中文翻译：
分数 mKdV 方程的修正散射
我们研究分数阶修正 Korteweg–de Vries 方程解的大时间渐近性
$$\begin{对齐} \left\{ \begin{array}{c} \partial _{t}u+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha -1}\partial _{x}u=\partial _{x}\left( u^{3}\right) ,~ t>0,\ x\in {\mathbb {R}}\ textbf{,} \\ u\left( 0,x\right) =u_{0}\left( x\right) ,\ x\in {\mathbb {R}}\textbf{,} \end{array} \正确的。\end{对齐}$$ (0.1)其中\(\alpha \in \left( 1,2\right) ,\) \(\left| \partial _{x}\right| ^{\alpha }={\mathcal {F}}^{-1 }\left| \xi \right| ^{\alpha }{\mathcal {F}}\)是分数阶导数。\(\alpha =3\)的情况对应于经典修正的KdV方程。在\(\alpha =2\)的情况下，它是修正的本杰明-小野方程。我们的目标是将 [10, 16] 中\(\alpha \in \left( 0,1\right) \) 的结果扩展到\ (\alpha \in \left( 1,2\right) \)。我们开发了基于因式分解技术的方法，该方法始于[11]，并将已知结果应用于\({\textbf{L}}^{2}\)- 伪微分算子的有界性以获得解的大时间渐近性。
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