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

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

1999-04-20

arxiv.org/abs/chao-dyn/9904036v1

Nonlinear Sciences

Chaotic Dynamics

30 pages

Scientific paper

10.1007/s002200050824

In this paper, one-dimensional (1D) nonlinear wave equations $u_{tt} -u_{xx}+V(x)u =f(u)$, 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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