Asymptotics of solutions for the fractional modified Korteweg-de Vries equation of order α ∈(2,3)

We continue to study the large time asymptotics of solutions for the fractional modified Korteweg-de Vries equation {partial derivative(t)u+1/alpha |partial derivative(x)|(alpha-1)partial derivative(x)u=partial derivative x(u(3)),t >0,x is an element of R, u(0,x)=u(0)(x), x is an element of R, where alpha is an element of(2,3),|partial derivative(x)|(alpha)=F-1|xi|F-alpha is the fractional derivative. This is a sequel to the previous works in which the cases alpha is an element of(0,1)boolean OR(1,2)were studied. It is known that the case of alpha=3 corresponds to the classical modified KdV equation. In the case of alpha=2 it is called the modified Benjamin-Ono equation. In the case alpha=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.The order alpha=2 is a critical point which divides the smoothing property and the derivative loss of solutions