Geodesic
The Cauchy problem for nonlinear Schrödinger-type equations
Sbornik. Mathematics, Tome 25 (1975) no. 3, pp. 429-440 
M. I. Vishik; A. V. Fursikov. The Cauchy problem for nonlinear Schrödinger-type equations. Sbornik. Mathematics, Tome 25 (1975) no. 3, pp. 429-440. http://geodesic.mathdoc.fr/item/SM_1975_25_3_a5/
@article{SM_1975_25_3_a5,
     author = {M. I. Vishik and A. V. Fursikov},
     title = {The {Cauchy} problem for nonlinear {Schr\"odinger-type} equations},
     journal = {Sbornik. Mathematics},
     pages = {429--440},
     year = {1975},
     volume = {25},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1975_25_3_a5/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The Cauchy problem for the equation $$ \frac{\partial u(t,x)}{\partial t}=-i\Delta u+f(u),\qquad f(u)=\sum_{k=2}^\infty f_ku^k, $$ is investigated. The solution is sought in the class of functions $u(t,x)$ which belong to a space $V_K'$ at each $t\in\mathbf R^1$. It is proved that the solution constructed is periodic in $t$. Bibliography: 2 titles.

[1] M. I. Vishik, A. V. Fursikov, “Nekotorye voprosy teorii nelineinykh ellipticheskikh i parabolicheskikh uravnenii”, Matem. sb., 94(136) (1974), 324–357

[2] M. I. Vishik, A. V. Fursikov, “Analiticheskie pervye integraly nelineinykh parabolicheskikh v smysle I. G. Petrovskogo sistem differentsialnykh uravnenii i ikh prilozheniya”, Uspekhi matem. nauk, XXIX:2(176) (1974), 123–153