Large time asymptotics for the fractional modified Korteweg-de Vries equation with $$\alpha \in \left( 2,4\right) $$ α ∈ 2 , 4

We study the large time asymptotics for solutions to the Cauchy problem for the fractional modified Korteweg-de Vries equation $$\begin{aligned} \left\{ \begin{array}{c} \partial _{t}w+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha -1}\partial _{x}w=\partial _{x}\left( w^{3}\right) ,\,t>0,\, x\in {\mathbb {R}}\mathbf {,} \\ w\left( 0,x\right) =w_{0}\left( x\right) ,\,x\in {\mathbb {R}}\mathbf {,} \end{array} \right. \end{aligned}$$ ∂ t w + 1 α ∂ x α - 1 ∂ x w = ∂ x w 3 , t > 0 , x ∈ R , w 0 , x = w 0 x , x ∈ R , where $$\alpha \in \left( 2,4\right) $$ α ∈ 2 , 4 , and $$\left| \partial _{x}\right| ^{\alpha }={\mathcal {F}}^{-1}\left| \xi \right| ^{\alpha }{\mathcal {F}}$$ ∂ x α = F - 1 ξ α F is the fractional derivative . The case of $$\alpha =3$$ α = 3 corresponds to the classical modified KdV equation. In the case of $$\alpha =2$$ α = 2 it is the modified Benjamin-Ono equation. Our aim is to find the large time asymptotic formulas for the solutions of the Cauchy problem for the fractional modified KdV equation. We develop the method based on the factorization techniques which was started in papers Hayashi, N., Naumkin, P.I. (Z. Angew. Math. Phys.) 59, 1002–1028 (2008), Hayashi, N., Naumkin, P.I. (SUT J. Math.) 52, 49–95 (2016) Hayashi, N., Ozawa, T.: (Ann. I.H.P. (Phys. Théor.)) 48, 17-37 (1988), Naumkin, P.I. (J. Differential Equations) 269(7), 5701–5729 (2020). Also we apply the known results on the $$\mathbf {L}^{2}$$ L 2 - boundedness of pseudodifferential operators.