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

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 Schr\"odinger 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 $\langle T \rangle \propto L^{-\gamma_a}$ and $\langle \ln T \rangle \approx -\gamma_g \ln L$ for sufficiently large $L$. At low input intensities, a transition from exponential to power-law decay in $\langle T \rangle$ occurs as $L$ increases. The exponents $\gamma_a$ and $\gamma_g$ are nearly identical, converging to approximately 0.5 as the strength of the nonlinear disorder, $\beta$, increases. Additionally, the variance of $T$ decays according to a power law with an exponent close to 1, and the variance of $\ln T$ 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 $\beta$. As $V$ increases, the region exhibiting power-law localization diminishes and eventually disappears when $V$ exceeds $\beta$, leading to standard Anderson localization.