Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 979–1010 | Cite as

A KAM Theorem for Higher Dimensional Forced Nonlinear Schrödinger Equations

  • Shuaishuai Xue
  • Jiansheng GengEmail author


In this paper, we prove an infinite dimensional KAM theorem. As an application, it is shown that there are many real-analytic small-amplitude linearly-stable quasi-periodic solutions for higher dimensional nonlinear Schrödinger equations with outer force
$$\begin{aligned} iu_t-\triangle u +M_\xi u+f(\bar{\omega }t)|u|^2u=0, \quad t\in {{\mathbb R}}, x\in {{\mathbb T}}^d \end{aligned}$$
where \(M_\xi \) is a real Fourier multiplier,\(f({\bar{\theta }})({\bar{\theta }}={\bar{\omega }} t)\) is real analytic and the forced frequencies \(\bar{\omega }\) are fixed Diophantine vectors.


Schrödinger equation Invariant torus Quasi-periodic solutions KAM theory 



We would like to thank the anonymous referee for many valuable suggestions.


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of MathematicsNanjing UniversityNanjingPeople’s Republic of China

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