Asymptotics for the fractional nonlinear Schrödinger equation with 2<α<52

We study the Cauchy problem for the fractional nonlinear Schrödinger equation i ∂ t u - 1 α ∂ x α u = λ u α u , t > 0 , x ∈ R , u 0 , x = u 0 x , x ∈ R , where λ ∈ R , the fractional derivative ∂ x α = F - 1 ξ α F , the order α ∈ 2 , 5 2 . Our aim is to find the asymptotics of solutions to the fractional nonlinear Schrödinger equation in the defocusing case λ > 0 . We show that the asymptotics differs from that in the case of the usual cubic nonlinear Schrödinger equation. To prove our main result, we develop the Factorization Techniques which was proposed in our previous works.