Consideration in this present paper is the long-time asymptotic of solutions to the derivative nonlinear Schr$\ddot{ o}$dinger equation with the step-like initial value \begin{ eqnarray} q(x,0)=q_{ 0}(x)=\begin{ cases} \begin{ split} A_{ 1}e^{ i\phi}e^{ 2iBx}, \quad\quad x<0,\\ A_{ 2}e^{ -2iBx}, \quad\quad~~ x>0. \end{ split}\nonumber \end{ cases} \end{ eqnarray} by Deift-Zhou method. The step-like initial problem described by a matrix Riemann-Hilbert problem. A crucial ingredient used in this paper is to introduce $g$-function mechanism for solving the problem of the entries of the jump matrix growing exponentially as $t\rightarrow\infty$. It is shown that the leading order term of the asymptotic solution of the DNLS equation expressed by the Theta function $\Theta$ about the Riemann-surface of genus 3 and the subleading order term expressed by parabolic cylinder and Airy functions.
Categories: nlin.SI