A self-consistent theory of localization in nonlinear random media

Abstract

The self-consistent theory of localization is generalized to account for a weak quadratic nonlinear potential in the wave equation. For spreading wave packets, the theory predicts the destruction of Anderson localization by the nonlinearity and its replacement by algebraic subdiffusion, while classical diffusion remains unaffected. In 3D, this leads to the emergence of a subdiffusion-diffusion transition in place of the Anderson transition. The accuracy and the limitations of the theory are discussed.

Prethermalization and wave condensation in a nonlinear disordered Floquet system

“…Analogous observations have been made in disordered nonlinear Schrödinger chains, see e.g. [57][58][59][60][61][62].…”

Coherent multiple scattering of out-of-equilibrium interacting Bose gases

“…Numerics nevertheless seems to suggest α = 0.3 − 0.4 in one dimension. The problem was also tackled in a random potential in three dimensions in the vicinity of the Anderson transition, where it was found that weak interactions also lead to sub-diffusion on the localized side of the transition, while they leave the critical point and the diffusive side of the transition essentially unaffected [85,86]. The question of wave-packet spreading is not fully clarified though, since other works also put forward that sub-diffusion could be non-algebraic and eventually become slower than any power law at arbitrarily long times [87,88].…”

Quench dynamics of a weakly interacting disordered Bose gas in momentum space

“…This approach was previously used in [35][36][37]39] to describe the stationary coherent backscattering effect of continuous waves in finite media, and in [48][49][50] to model the dynamics of interacting wave packets in the diffusive limit. The latter configuration was later extended to the localization regime in [22,23], but by taking into account first-order corrections in g only, see Sec. IV B, while neglecting second-order corrections responsible for thermalization.…”

“…In weakly interacting Bose gases, which can be described at a mean-field level with a nonlinear Schrödinger equation, many-body localization does not occur and thermalization is the rule. Notwithstanding the relative simplicity of this limit, the combination of weak interactions and disorder still gives rise to a number of puzzling phenomena, such as thermalization via weakly coupled localized states [17,18] or subdiffusive spreading of wave packets [19][20][21][22][23][24].…”