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KAM for the nonlinear Schrödinger equation. (English) Zbl 1201.35177
The authors consider a complex, multidimensional NLS equation in the analytic and periodic framework. The linear equation possesses quasiperiodic solutions with linear frequencies \(\omega_a=|a|^2+\hat V(a)\), with \(a\) integer and \(\widehat V\) the Fourier coefficient of the potential \(V\).
The authors use such frequencies as external parameters that somewhat modulate the normal frequencies of the system, in order to overcome the difficulties arising from the strong degeneracy and the infinitely many degrees of freedom of the system. In this way, they are able to prove a KAM Theorem, showing the persistence of the quasiperiodic solutions for a large set of frequencies. The Toeplitz-Lipschitz property and the reducibility are important technical ingredients of the proof, in which infinitely many arithmetic conditions on the small divisors have to be addressed simultaneously.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
70H08 Nearly integrable Hamiltonian systems, KAM theory
81Q99 General mathematical topics and methods in quantum theory
35B09 Positive solutions to PDEs
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