Large time asymptotics for the fractional modified Korteweg-de Vries equation of order $$\alpha \in \left[ 4,5\right) $$

Abstract We study the large time asymptotics of solutions to Cauchy problem for fractional modified Korteweg-de Vries equation $$\begin{aligned} \left\{ \begin{array}{l} \partial _{t}w+\frac{1}{\alpha }\left| _{x}\right| ^{\alpha -1}\partial _{x}w=\partial _{x}\left( w^{3}\right) ,\text { }t>0,\, x\in {\mathbb {R}}\textbf{,}\\ w\left( 0,x\right) =w_{0}\left( x\right) ,\,x\in {R}} \textbf{,} \end{array} \right. \end{aligned}$$ ∂ t w + 1 α x - = 3 , > 0 ∈ R where $$\alpha \in \left[ 4,5\right) ,$$ 4 5 and $$\left| }={\mathcal {F}}^{-1}\left| \xi \right| }{\mathcal {F}}$$ F ξ is derivative. The case =3$$ corresponds classical KdV equation. In =2$$ 2 it Benjamin–Ono Our aim find asymptotic formulas develop method based on factorization techniques which was started in our previous papers. Also we apply known results $${\textbf{L}}^{2}$$ L —boundedness pseudodifferential operators.