Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces

The local well-posedness with small data in Hs(Rn)(s⩾3+max⁡(n/2,1+)) for the Cauchy problem of the fourth order nonlinear Schrödinger equations with the third order derivative nonlinear terms were obtained by Huo and Jia [17]. In this paper we show its global well-posedness with small data in the modulation space View the MathML source and in Sobolev spaces Hn/2+7+/2. For a special nonlinear term containing only one third order derivative, we can show its global well posedness in View the MathML source and H(n+1+)/2