We derive the time evolution of the two-mode amplitude probability density function. Using this
equation, we derive conditions for the existence of a zero flux steady-state solution. We also derive the
equation for a vortex solution and show that the product of two one-mode steady-state solutions can be a
two-mode steady-state solution only when an extra condition is satisfied. With this extra condition
assumed, we plot the flux of probability vector on two mode’s plane. It is shown that this flux lines
circulate around the center (n_a, n_b), which are the mean values of the two mode’s amplitude square.