Abstract:
The nonlinear equation
\begin{equation}
i\frac{du}{dt}=(\alpha-\beta i)u_{xx}+\gamma u+\sum_{k=2}^\infty\varphi_ku^k
\end{equation}
on the real axis is reduced (for $\alpha$, $\beta$, $\gamma$ real, $\beta\ne0$,
$\gamma\ne 0$) by a differentiable change of variables in a neighborhoodd of zero of the Banach space $U$ to the linear equation
\begin{equation}
i\frac{dv}{dt}=(\alpha-i\beta)v_{xx}+\gamma v.
\end{equation}