Renato Calleja
Existence and persistence of invariant objects in dynamical systems and mathematical physics
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ABSTRACT. In this dissertation we present four papers as chapters.
In Chapter 2, we extended the
techniques used for the Klein-Gordon Chain by
Iooss, Kirchg\"assner, James, and Sire, to chains with
non-nearest neighbor interactions.
We look for travelling waves by reducing
the Klein-Gordon chain with second nearest neighbor interaction
to an
advance-delay equation.
Then we reduce the equation
to a finite dimensional
center manifold for some parameter regimes.
By using the normal form expansion on the center
manifold we were able to prove the existence of three
different types of travelling solutions for the Klein
Gordon Chain: periodic, quasi-periodic and homoclinic to
periodic orbits with exponentially small amplitude.
In Chapter 3 we include numerical methods for computing quasi-periodic solutions.
We developed very efficient algorithms
to compute smooth quasi-periodic equilibrium states of
models in 1-D statistical mechanics models allowing non-nearest neighbor interactions.
If we discretize a hull function using $N$ Fourier coefficients,
the algorithms
require $O(N)$ storage and a Newton step for the
equilibrium equation requires only
$O(N \log(N))$ arithmetic operations.
This numerical methods give rise to a criterion for the
breakdown of quasi-periodic solutions.
This criterion is presented in Chapter 4.
In Chapter 5, we justify rigorously the criterion in Chapter 4.
The justification of the criterion uses both
Numerical KAM algorithms and rigorous results.
The hypotheses of the theorem concern bounds on the Sobolev norms of
a hull function and can be verified
rigorously by the computer. The argument works with small modifications in
all cases where there is an \emph{a posteriori} KAM theorem.