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The interplay of fluctuations, ergodicity, and disorder in many-bodyinteracting systems has been striking attention for half a century, pivoted ontwo celebrated phenomena: Anderson localization predicted in disordered media,and Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence observed in a nonlinear system.The destruction of Anderson localization by nonlinearity and the recovery ofergodicity after long enough computational times lead to more questions. Thisthesis is devoted to contributing to the insight of the nonlinear systemdynamics in and out of equilibrium. Focusing mainly on the GP lattice, weinvestigated elementary fluctuations close to zero temperature, localizationproperties, the chaotic subdiffusive regimes, and the non-equipartition ofenergy in non-Gibbs regime.  Initially, we probe equilibrium dynamics in the ordered GP lattice and reporta weakly non-ergodic dynamics, and an ergodic part in the non-Gibbs phase thatimplies the Gibbs distribution should be modified. Next, we include disorder inGP lattice, and build analytical expressions for the thermodynamic propertiesof the ground state, and identify a Lifshits glass regime where disorderdominates over the interactions. In the opposite strong interaction regime, weinvestigate the elementary excitations above the ground state and found adramatic increase of the localization length of Bogoliubov modes (BM) withincreasing particle density. Finally, we study non-equilibrium dynamics withdisordered GP lattice by performing novel energy and norm density resolved wavepacket spreading. In particular, we observed strong chaos spreading overseveral decades, and identified a Lifshits phase which shows a significantslowing down of sub-diffusive spreading.

Equilibrium and Non-equilibrium Gross-Pitaevskii Lattice Dynamics: Interactions, Disorder, and Thermalization