Long-time asymptotic behavior for the Hermitian symmetric space derivative nonlinear Schrödinger equation

Resorting to the spectral analysis of the 4 x 4 matrix spectral problem, we construct a 4 x 4 matrix Riemann-Hilbert problem to solve the initial value problem for the Hermitian symmetric space derivative nonlinear Schr & ouml;dinger equation. The nonlinear steepest decent method is extended to study the 4 x 4 matrix Riemann-Hilbert problem, from which the various Deift-Zhou contour deformations and the motivation behind them are given. Through some proper transformations between the corresponding Riemann-Hilbert problems, the basic Riemann-Hilbert problem is reduced to a model Riemann-Hilbert problem, by which the long-time asymptotic behavior to the solution of the initial value problem for the Hermitian symmetric space derivative nonlinear Schr & ouml;dinger equation is obtained with the help of the asymptotic expansion of the parabolic cylinder function and strict error estimates.