Long-time Asymptotic for the Derivative Nonlinear Schrodinger Equation with Step-like Initial Value

We study long-time asymptotics of the solution to the Cauchy problem for the Gerdjikov-Ivanov type derivative nonlinear Schrodinger equation iq(t) + q(xx) - iq(2)(q) over bar (x) + 1/2 vertical bar q vertical bar(4)q = 0 with step-like initial data q(x, 0) = 0 for x <= 0 and q(x, 0) = Ae(-2i Bx) for x > 0, where A > 0 and B is an element of R are constants. We show that there are three regions in the half-plane {(x, t)| - infinity < x < infinity, t > 0}, on which the asymptotics has qualitatively different forms: a slowly decaying self-similar wave of Zakharov-Manakov type for , a plane wave region: x < -4t (B + root 2A(2) (B + A(2)/4)), an elliptic region: -4t (B + root 2A(2) (B + A(2)/4)) < x < -4tB. Our main tools include asymptotic analysis, matrix Riemann-Hilbert problem and Deift-Zhou steepest descent method.