Geodesic
On the normal form of nonlinear partial differential equations on the real axis
Sbornik. Mathematics, Tome 34 (1978) no. 1, pp. 111-117 Cet article a éte moissonné depuis la source Math-Net.Ru

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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} Bibliography: 3 titles. 
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V. I. Sedenko. On the normal form of nonlinear partial differential equations on the real axis. Sbornik. Mathematics, Tome 34 (1978) no. 1, pp. 111-117. http://geodesic.mathdoc.fr/item/SM_1978_34_1_a4/

[1] B. A. Plise, “O privedenii analiticheskoi sistemy differentsialnykh uravnenii k lineinoi forme”, Diff. uravneniya, 1:2 (1965), 153–161

[2] N. V. Nikolenko, “Polnaya integriruemost nelineinogo uravneniya Shredingera”, DAN SSSR, 227:4 (1976), 235–238 | MR

[3] A. N. Kolmogorov, “O sokhranenii uslovno-periodicheskikh dvizhenii pri malom izmenenii funktsii Gamiltona”, DAN SSSR, 98:4 (1954), 527–530 | MR | Zbl