Solitonization of the Anderson Localization

Solitons and Anderson localization have several features in common: exponential localization, negative eigenvalues, criticality in two dimensions, possibility of being located anywhere in space. However, at first glance, they are two completely different forms of wave localization, solitons being due to nonlinearity and Anderson states due to a linear disordered potential.

Hence, even if the mentioned affinities are evident, one could be tempted to exclude any connection. But one also could not.

In a paper on the arXiv, [Phys. Rev. A 86, 016801(R) (2012)] C. Conti reports on a theoretical analysis in one-dimension showing that a properly defined disorder averaged equation allows to derive closed form expressions for the shape of fundamental Anderson state and its features (eigenvalue and localization length) in the presence of nonlinearity. Such an equation is a nonlinear Schroedinger equations, the very same sustaining solitons.

In the picture below, the numerically calculated profiles of the nonlinear Anderson states are shown, and the fact that their shape resembles a bright soliton when increasing nonlinearity (the power P) evidenced.