  In this thesis, we present a comprehensive study of chaos and thermalization
of the one-dimensional Bose-Hubbard Model (BHM) within the classical field
approximation. Two quantitative measures are compared: the ensemble-averaged
Finite-time Maximal Lyapunov exponent, a measures of chaos and the normalized
spectral entropy, a measure of the distance between the numerical time-averaged
momentum distribution and the one predicted by thermodynamics. A threshold for
chaos is found, which depends on two parameters, the nonlinearity and the total
energy-per-particle. Below the threshold, the dynamics are regular, while far
above the threshold, complete thermalization is observed, as measured by the
normalized spectral entropy. We study individual resonances in the Bose-Hubbard
model to determine the criterion for chaos. The criterion based on Chirikov's
method of overlapping resonances diverges in the thermodynamic limit, in
contrast to the criterion parameters inferred from numerical calculations,
signifying the failure of the standard Chirikov's approach. The Ablowitz-Ladik
lattice is one of several integrable models that are close to the BHM. We
outline the method of Inverse Scattering Transform and generate the integrals
of motion of the Ablowitz-Ladik lattice. Furthermore, we discuss the possible
role of these quantities in the relaxation dynamics of the BHM.
