On a Weak Solution of the Mixed Nonlinear Schrödinger Equations

In this paper we discuss the mixed nonlinear Schrödinger Equation Ut = iαUxx + βU2Ux + γ|U|2Ux + ig(|U|2)U, (∗) U(x, 0) = U0(x), where α, β, γ are real constants with α ≠ 0. If the real function g(s) ∈ C2(R+) satisfies some conditions, and the initial data U0(x) ∈ H1(R) are such that the norm ||U0||L2(R) is sufficiently small, then there is a unique weak solution U(x, t) of (∗) such that U(x, t) ∈ Cw(R; H1(R)).