12-26 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) Feb 17, 12
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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.

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