Asymptotic behavior for solutions to the vector nonlinear Schrödinger equations

Abstract

In this paper, we study the Cauchy problem for the vector nonlinear Schrödinger equations with cubic nonlinearities. The vector nonlinear Schrödinger equations are also called as Manakov systems. In particular, our main purpose of this study is to consider the asymptotic behavior of global solutions and our main results of this study shows the existence of asymptotically free solutions for space dimensions n=2,3 in the intersection of Sobolev space and weighted Lebesgue space Hs∩FHγ with s,γ∈(0,1]∩(n/2−1,n/2) and existence of modified asymptotically free solutions for space dimension n=1 in the weighted Lebesgue space FHγ with 1/2<γ≤1.