In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain in [15] on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions. The persistence result is given through a rather abstract “counter-term theorem” à la Herman, directly in the original elliptic variables without passing to action-angle ones. Our framework allows us to find “many more” almost periodic solutions with respect to the existing literature and consider also non-translation invariant PDEs.

在本文中，我们研究了带有外部参数的圆上的NLS方程的概周期解的存在性和线性稳定性。从Bourgain在[15]中对五次NLS的开创性结果开始，我们提出了一种新颖的方法，可以在统一的框架中证明有限维和无限维不变托里的持久性，这是所需解决方案的支持。持久性结果是通过一个相当抽象的“逆定理”àla Herman给出的，直接在原始椭圆形变量中给出，而没有传递给动作角变量。我们的框架使我们能够针对现有文献找到“更多”几乎周期性的解决方案，并考虑非翻译不变的PDE。