In this paper we discuss the mixed nonlinear Schrodinger Equation Ut=iαUxx+βU2Ux+γ|U|2Ux+ig(|U|2) U, U (x, 0)=Uo(x), where x, β,γ are constants with α≠0. If the real function g(s)εC2(R+) satisfies some conditions, and the initial data U0(x)εH1 (R) are such that the norm |U0|L2(R) is sufficiently small. then there is a unique weak soultion U (x, 1) of (*) such that U (x, 1)εCw(R; H1 (R)).