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

TL;DR

This paper investigates the critical transition between localization and delocalization in a disordered nonlinear Schrödinger system, revealing subdiffusive spreading characterized by specific power-law exponents and establishing a connection with percolation and dynamical systems.

Contribution

It demonstrates that the localization-delocalization transition is a critical phenomenon with a precise power-law spreading exponent and provides an exact threshold for unlimited spreading using a Cayley tree mapping.

Findings

Critical spreading exponent is 1/3 near the transition.

Above the critical point, spreading is subdiffusive with exponent 2/5.

An exact threshold for unlimited spreading is derived.

Abstract

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 $t\rightarrow+\infty$. 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 $\alpha = 2/5$ consistently with the results from previous investigations. A threshold for unlimited spreading is calculated exactly by mapping the transport problem on a Cayley tree.