Dynamic chaos in the solution of the Gross-Pitaevskii equation for a periodic potential

We analytically and numerically investigate the solution to the stationary Gross-Pitaevskii equation for a one-dimensional potential of the optical lattice in the case of repulsive nonlinearity. From the mathematical viewpoint, this problem is similar to the well-known problem of the classical mathematical Kapitza pendulum perturbed by a weak high-frequency force. At certain values of the parameters, dynamic chaos is produced in the considered problem. It is modeled analytically by a nonlinear diffusion equation.