Modified scattering for the fractional mKdV equation

Abstract

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.