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KAM Tori for 1D Nonlinear Wave Equations¶with Periodic Boundary Conditions

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In this paper, one-dimensional (1D) nonlinear wave equations

with periodic boundary conditions are considered; V is a periodic smooth or analytic function and the nonlinearity f is an analytic function vanishing together with its derivative at u≡0. It is proved that for “most” potentials V(x), the above equation admits small-amplitude periodic or quasi-periodic solutions corresponding to finite dimensional invariant tori for an associated infinite dimensional dynamical system. The proof is based on an infinite dimensional KAM theorem which allows for multiple normal frequencies.

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Received: 2 August 1999 / Accepted: 7 January 2000

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Chierchia, L., You, J. KAM Tori for 1D Nonlinear Wave Equations¶with Periodic Boundary Conditions. Comm Math Phys 211, 497–525 (2000).

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