On the normal form of nonlinear partial differential equations on the real axis

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}

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