Global well-posedness and scattering for the fourth order nonlinear Schrödinger equations with small data in modulation and Sobolev spaces (Q897832)

The local well-posedness with small data in \(H^s(\mathbb R^n)~(s\geq 3 + \max(n/2, 1_+))\) for the Cauchy problem of the fourth order nonlinear Schrödinger equations with the third order derivative nonlinear terms were obtained by \textit{Z. Huo} and \textit{Y. Jia} [J. Math. Pures Appl. (9) 96, No. 2, 190--206 (2011; Zbl 1231.35234)]. In this paper the authors show its global well-posedness with small data in the modulation space \(M^{7/2}_{2,1}\) and in Sobolev spaces \(H^{n/2+7_+/2}\). For a special nonlinear term containing only one third order derivative, the authors can show its global well posedness in \(M^{1/2}_{2,1}\) and \(H^{(n+1_+)/2}\).