  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.
