Nonlinear dynamical phase diagram of ultra-cold atoms in periodic lattices

We study the dynamics of ultra-cold bosonic gases in periodic deep optical potential by incorporating the nonlinear inter-site coupling into the discrete nonlinear Schr\"{O}dinger equation (DNLSE). We numerically solve the DNLSE to analyze and compare the evolution of Gaussian wave packets reported in [Phys. Rev. Lett., 86, 11(2001)]. Our result corroborates the validity of the phase diagram calculated by the variational method within a certain range of parameters of the initial wave packets. We calculate the long-term evolution of solutions of DNLSE to generate a fine phase diagram to elaborate the transitions between different dynamic modes. We propose an experimental scheme to verify the phase diagram based on the Bose-Einstein condensates in optical lattices.