Power-law localization in one-dimensional systems with nonlinear disorder under fixed input conditions

Abstract

We conduct a numerical investigation into wave propagation and localization in one-dimensional lattices subject to nonlinear disorder, focusing on cases with fixed input conditions. Utilizing a discrete nonlinear Schrödinger equation with Kerr-type nonlinearity and a random coefficient, we compute the averages and variances of the transmittance, T, and its logarithm, as functions of the system size L, while maintaining constant intensity for the incident wave. In cases of purely nonlinear disorder, we observe power-law localization characterized by 〈T〉∝L−γa and 〈lnT〉≈−γglnL for sufficiently large L. At low input intensities, a transition from exponential to power-law decay in 〈T〉 occurs as L increases. The exponents γa and γg are nearly identical, converging to approximately 0.5 as the strength of the nonlinear disorder, β, increases. Additionally, the variance of T decays according to a power law with an exponent close to 1, and the variance of lnT approaches a small constant as L increases. These findings are consistent with an underlying log-normal distribution of T and suggest that wave propagation behavior becomes nearly deterministic as the system size increases. When both linear and nonlinear disorders are present, we observe a transition from power-law to exponential decay in transmittance with increasing L when the strength of linear disorder, V, is less than β. As V increases, the region exhibiting power-law localization diminishes and eventually disappears when V exceeds β, leading to standard Anderson localization.