Ernest Fontich, Rafael de la Llave, Yannick Sire
Construction of invariant whiskered tori by a
parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices.
(982K, pdf)
ABSTRACT. We construct quasi-periodic and almost periodic solutions
for coupled Hamiltonian systems on an infinite lattice which are translation
invariant. The couplings can be
long range, provided that they decay moderately fast with
respect to
the distance.
For the solutions we construct, most of the sites are moving
in a neighborhood of
a hyperbolic fixed point, but there are oscillating sites clustered
around a sequence of nodes. The amplitude of these oscillations
does not need to tend to zero. In particular, the almost periodic solutions
do not decay at infinity.
The main result is an a-posteriori theorem.
We formulate an invariance equation. Solutions of
these equation are embeddings of an invariant torus.
We show that, if there is an approximate solution of
the invariance equation that satisfies some non-degeneracy
conditions, there is a true solution close by.
This
does not require that the system
is close to integrable, hence it can be used to
validate numerical
calculations or formal expansions.
The proof of this a-posteriori theorem is based on a
Nash-Moser iteration, which does not use transformation theory.
Simpler versions of the scheme were developed in
E. Fontich, R. de la Llave,Y. Sire \emph{Jour. Diff. Eq.} {f 246},
3136 (2009).
One technical tool, important for our purposes, is the use of
weighted spaces that capture the idea that the maps under consideration
are local interactions. Using these weighted spaces,
the estimates of
iterative steps are similar to those in finite dimensional
spaces. In particular, the estimates are independent of the number of
nodes that get excited.
Using these techniques, given two breathers, we can place them
apart and obtain an approximate solution, which leads to
a true solution nearby. By repeating the process infinitely often,
we can get solutions with infinitely many frequencies which
do not tend to zero at infinity.