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

Abstract

We study the large time asymptotics of solutions to the Cauchy problem for the fractional modified Korteweg-de Vries equation ∂tw+1α∂xα-1∂xw=∂xw3,t>0,x∈R,w0,x=w0x,x∈R,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\left\\{ \\begin{array}{l} \\partial _{t}w+\\frac{1}{\\alpha }\\left| \\partial _{x}\\right| ^{\\alpha -1}\\partial _{x}w=\\partial _{x}\\left( w^{3}\\right) ,\ ext { }t>0,\\, x\\in {\\mathbb {R}}\ extbf{,}\\\\ w\\left( 0,x\\right) =w_{0}\\left( x\\right) ,\\,x\\in {\\mathbb {R}} \ extbf{,} \\end{array} \\right. \\end{aligned}$$\\end{document}where alpha in left[ 4,5right) , and 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 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 our previous papers. Also we apply the known results on the {textbf{L}}^{2}—boundedness of pseudodifferential operators.