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In this thesis, we present a comprehensive study of chaos and thermalizationof the one-dimensional Bose-Hubbard Model (BHM) within the classical fieldapproximation. Two quantitative measures are compared: the ensemble-averagedFinite-time Maximal Lyapunov exponent, a measures of chaos and the normalizedspectral entropy, a measure of the distance between the numerical time-averagedmomentum distribution and the one predicted by thermodynamics. A threshold forchaos is found, which depends on two parameters, the nonlinearity and the totalenergy-per-particle. Below the threshold, the dynamics are regular, while farabove the threshold, complete thermalization is observed, as measured by thenormalized spectral entropy. We study individual resonances in the Bose-Hubbardmodel to determine the criterion for chaos. The criterion based on Chirikov'smethod of overlapping resonances diverges in the thermodynamic limit, incontrast to the criterion parameters inferred from numerical calculations,signifying the failure of the standard Chirikov's approach. The Ablowitz-Ladiklattice is one of several integrable models that are close to the BHM. Weoutline the method of Inverse Scattering Transform and generate the integralsof motion of the Ablowitz-Ladik lattice. Furthermore, we discuss the possiblerole of these quantities in the relaxation dynamics of the BHM.

Chaos and Thermalization in the one-dimensional Bose-Hubbard model in the classical-field approximation