$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\Aex}{{A_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
$$
Extension to a variable coefficient; operator notation
$$
\begin{align*}
\lbrack D^+_t u &= -au\rbrack^n,\\
\lbrack D^-_t u &= -au\rbrack^n,\\
\lbrack D_t u &= -a\overline{u}^t\rbrack^{n+\half},\\
\lbrack D_t u &= -\overline{au}^t\rbrack^{n+\half}\\
\end{align*}
$$
