The Cauchy problem for nonlinear Schrödinger equations in modulation spaces

We consider the nonlinear Schrödinger equation equation(∗∗){iut+Δu±f(u)=0,f(u)=|u|2mun,m,n∈N,(x,t)∈RN×Ru(0,x)=u0(x). We give a representation of the unique solution of the Cauchy problem (∗∗) existed in C(0,T∗;Mp,1)C(0,T∗;Mp,1). Moreover, by this representation, we obtain that there exists a constant B independent of p,Np,N such that for any initial data ‖u0‖Mp′,12N>2, the Schrödinger equation (∗∗) has a unique global solution u∈C(−∞,∞;Mp,1)u∈C(−∞,∞;Mp,1).