Asymptotics of solutions for the fractional modified Korteweg–de Vries equation of order $$\alpha \in \left( 2,3\right) $$ α ∈ 2, 3

Rafael Carreño-Bolaños (), Nakao Hayashi () and Pavel I. Naumkin ()
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Rafael Carreño-Bolaños: Instituto Tecnológico de Morelia
Nakao Hayashi: Tohoku University
Pavel I. Naumkin: UNAM Campus Morelia

Partial Differential Equations and Applications, 2023, vol. 4, issue 4, 1-15

Abstract: Abstract We continue to study the large time asymptotics of solutions for the fractional modified Korteweg–de Vries equation $$\begin{aligned} \left\{ \begin{array}{ll} \partial _{t}u+\frac{1}{\alpha }\left| \partial _{x}\right| ^{\alpha -1}\partial _{x}u=\partial _{x}\left( u^{3}\right) ,&{}\quad t>0,\ x\in {\mathbb {R}}, \\ u\left( 0,x\right) =u_{0}\left( x\right) ,&{}\quad x\in {\mathbb {R}}, \end{array} \right. \end{aligned}$$ ∂ t u + 1 α ∂ x α - 1 ∂ x u = ∂ x u 3 , t > 0 , x ∈ R , u 0 , x = u 0 x , x ∈ R , where $$\alpha \in \left( 2,3\right) ,$$ α ∈ 2 , 3 , $$\left| \partial _{x}\right| ^{\alpha }={\mathcal {F}}^{-1}\left| \xi \right| ^{\alpha }{\mathcal {F}}$$ ∂ x α = F - 1 ξ α F is the fractional derivative. This is a sequel to the previous works in which the cases $$\alpha \in \left( 0,1\right) \cup \left( 1,2\right) $$ α ∈ 0 , 1 ∪ 1 , 2 were studied. It is known that the case of $$\alpha =3$$ α = 3 corresponds to the classical modified KdV equation. In the case of $$\alpha =2$$ α = 2 it is called the modified Benjamin–Ono equation. In the case $$\alpha =1,$$ α = 1 , it is the nonlinear wave equation and the exceptional case. Our aim is to find the large time asymptotic formulas of solutions. Main difference between the previous works and our result is in the order of fractional derivative $$\alpha .$$ α . The order $$\alpha =2$$ α = 2 is a critical point which divides the smoothing property and the derivative loss of solutions.

Keywords: Fractional mKdV equation; Modified scattering; Large time symptotics; 35B40; 35Q92 (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s42985-023-00247-x

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