The Cauchy problem for nonlinear Schrödinger-type equations

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 $t$.
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