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

¹ Associazione EURATOM-ENEA sulla Fusione, C. R. Frascati - Via E. Fermi 45, C.P.-65, I-00044 Frascati, Rome, Italy, EU
² Department of Physics and Solid State Institute, Technion-Israel Institute of Technology - Haifa, 32000, Israel

Received: 15 March 2012
Accepted: 16 September 2012

Abstract

Localization-delocalization transition in a discrete Anderson nonlinear Schrödinger 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 . The second moment grows with time as a power law ∝t^α, with α exactly 1/3. This critical spreading finds its significance in association 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 α = 2/5 consistently with the results from previous investigations. A threshold for unlimited spreading is calculated exactly by mapping the transport problem on a Cayley tree.