On the derivative nonlinear Schrödinger equation

In this paper we discuss the Cauchy problem for the derivative nonlinear Schrödinger equation: i∂_tψ + 2iδ∂_x(|;ψ|²ψ) = 0, ψ(0, x) = f{cyrillic}(x), where δ ≠ 0. Under an explicit smallness condition of the initial data, we prove the unique global existence of solutions to this problem in the usual Sobolev spaces, in the weighted Sobolev spaces, and in the Schwartz class. We describe the smoothing effect in detail. Furthermore, for the data decaying exponentially at infinity we prove that the above equation has unique local solutions which are analytic in the space direction.