Localization-delocalization transition on a separatrix system of nonlinear Schrodinger equation with disorder

Localization-delocalization transition in a discrete Anderson nonlinear Schr"odinger equation with disorder is shown to be a critical phenomenon $-$ similar to a percolation transition on a disordered lattice, with the nonlinearity parameter thought as the control parameter. In vicinity of the critical point the spreading of the wave field is subdiffusive in the limit $t\to +\mathrm{\infty }$. The second moment grows with time as a powerlaw $\propto t^\alpha$,with$\alpha$ exactly 1/3. This critical spreading finds its significance in some connection with the general problem of transport along separatrices of dynamical systems with many degrees of freedom and is mathematically related with a description in terms fractional derivative equations. Above the delocalization point, with the criticality effects stepping aside, we find that the transport is subdiffusive with $\alpha =2\mathrm{/}5$ consistently with the results from previous investigations. A threshold for unlimited spreading is calculated exactly by mapping the transport problem on a Cayley tree.