In this paper, an infinite dimensional KAM theorem with double normal frequencies is established under qualitative non-degenerate conditions. This is an extension of the degenerate KAM theorem with simple normal frequencies in [3] by Bambusi, Berti and Magistrelli. As applications, for nonlinear wave equation and nonlinear Schr$ \ddot{\mbox{o}} $dinger equation with periodic boundary conditions, quasi-periodic solutions of small amplitude and quasi-periodic solutions around plane wave are obtained respectively.

中文翻译：

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具有周期边界条件的偏微分方程的退化KAM定理

在定性非退化条件下，建立了具有双正常频率的无穷维KAM定理。这是简并KAM定理的扩展，具有[3]由Bambusi，Berti和Magistrelli撰写。作为应用，对于具有周期性边界条件的非线性波动方程和非线性Schr $ \ ddot {\ mbox {o}} $ dinger方程，分别获得了小振幅的准周期解和围绕平面波的准周期解。