A quadratic nonlinear Schrödinger equation in one space dimension

In this paper, we study the Cauchy problem for the quadratic derivative nonlinear Schrödinger equation We suppose that the initial data are small in the weighted Sobolev space H3,1(R) then we prove that there exist global in time small solutions to the Cauchy problem (*). We study the large time behavior of solutions and construct the modified asymptotics for large time values