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

Abstract

In this paper, one-dimensional (1D) nonlinear wave equationswith 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. * The research was partly supported by NNSF of China and by the italian CNR grant # 211.01.31. 1 s ( F z 0 Da,ρ(r,s),O + n∈Z + F zn Da,ρ(r,s),O nāe nρ ). Notational Remark In what follows, only the indices r, s and the set O will change while a,ā, ρ will be kept fixed, therefore we shall usually denote X F ā,ρ Da,ρ(r,s),O by X F r,s,O ,D a,ρ (r, s) by D(r, s) and F Da,ρ(r,s),O by F r,s,O . 3 Dot stands for the time derivatives d/dt. 4 The norm · Da,ρ(r,s),O for scalar functions is defined in (2.3). For vector (or matrix-valued) functions G : Da,ρ(r, s) × O → C m , (m < ∞) is similarly defined as G Da,ρ(r,s),O = m i=1 Gi Da,ρ(r,s),O (for the matrix-valued case the sum will run over all entries). 14 Recall the definition of Pi in (4.6). 15 Recall the definition of N in (4.1).

Cantor families of periodic solutions for wave equations via a variational principle

“…This difficulty was first solved by Kuksin [21] and Wayne [29] using KAM theory (other existence results of quasi-periodic solutions via KAM theory have been obtained in [23,25,26,12,22,17]). …”

A KAM theorem for one dimensional Schrödinger equation with periodic boundary conditions

“…Essentially, KAM machinery includes two parts: analytic part which deals with the iteration and proves convergence under some small divisor conditions, and geometric part which proves that the parameter set left after infinitely many times iteration has positive Lebesgue measure. In [7], Chierchia and You improved the analytic part of the KAM machinery so that it applies to multiple normal frequency case encountered in 1D Hamiltonian PDEs with periodic boundary conditions. For the geometric part, one has to assume a kind of regularity property, i.e., the vector field generated by nonlinear terms of the PDEs sends a sequence with decay to a sequence with faster decay to guarantee that there are essentially finitely many resonances at each KAM step (for truncated K).…”

“…Without loss of generality, we assume | k, + n | 1 and | k, ± n ± m | 1. For the proofs of the above lemmata, we refer the readers to the Appendix in [7]. …”

Periodic Solutions for a Class of Nonlinear Partial Differential Equations in Higher Dimension

“…We write 9) and introduce the coefficients u ± ν by setting u + ν := u ν and u − ν := u ν . Analogously we define 10) and in the case of Dirichlet boundary conditions we shall require u ν = −u Si(ν) for all i = 1, . .…”