Long-time Asymptotic Behavior for the Derivative Schrodinger Equation with Finite Density Type Initial Data

In this paper, the authors apply partial derivative steepest descent method to study the Cauchy problem for the derivative nonlinear Schrodinger equation with finite density type initial data iq(t) + q(xx) + i(vertical bar q vertical bar(2)q)(x) = 0, q(x, 0) = q(0)(x), where lim(x ->+/-infinity) q(0) (x) = q +/- and vertical bar q +/-vertical bar = 1. Based on the spectral analysis of the Lax pair, they express the solution of the derivative Schrodinger equation in terms of solutions of a Riemann-Hilbert problem. They compute the long time asymptotic expansion of the solution q(x,t) in different space-time regions. For the region xi = x/t with vertical bar xi + 2 vertical bar < 1, the long time asymptotic is given by q(x, t) = T(infinity)(-2)q(Lambda)(r)(x, t) + O(t(-3/4)), in which the leading term is N(I) solitons, the second term is a residual error from a partial derivative equation. For the region vertical bar xi + 2 vertical bar > 1, the long time asymptotic is given by q(x, t) = T(infinity)(-2)q(Lambda)(r)(x, t) - t(-1/2) if(11) + O(t(-3/4)), in which the leading term is N(I) solitons, the second t(-1/2) order term is soliton-radiation interactions and the third term is a residual error from a partial derivative equation. These results are verification of the soliton resolution conjecture for the derivative Schrodinger equation. In their case of finite density type initial data, the phase function theta(z) is more complicated that in finite mass initial data. Moreover, two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis, respectively.