Underpinning research on the dynamical aspects of one-dimensional nonlinear lattices

The main goal of this thesis is to provide an atlas of methodologies to tackle discrete systems containing inborn nonlinear contributions. Studies of lattices date back to the late 1800s when scientists of the level of William Thomson (a.k.a. Lord Kelvin), L'eon Brillouin and many others started exploring models of dispersion. Nowadays, applications are widely extended to an uncountable number of research fields, from physics to mathematics, but also engineering and chemistry. There is a trace of nonlinearity everywhere because our world is inherently chaotic, stochastic, and, despite many people not liking the idea, uncontrollable to a certain extent. This research focuses on 1-dimensional strips, namely selected structures composed of masses and springs. The high complexity of the study has its origin in two aspects: the dynamical equations coupling and the nonlinear forcing. In their linear approximation, solutions can be (somewhat trivially depending on the architecture of the problem) obtained as superimposition of decoupled harmonic functions. When the system is perturbed by nonlinear forcing, an analytical solution might not even exist. Our idea takes its shape and finds fertile ground in the celebrated Fermi-Pasta-Ulam-Tsingou chain, or as it is better known, the FPU"T" paradox (acronym derived from the scientists who designed it: Fermi, Pasta, Ulam, and only recently recognised contribution of Marie Tsingou. Note that most of the literature up to 2010 still makes use of the shorter version FPU). The problem is attacked both mathematically and numerically, showing the weaknesses and strengths of several approaches proposed during the last 70 years and proposing new techniques which act as glue between the antipodal dynamical features that the model displays.