Power-law localization in one-dimensional systems with nonlinear disorder under fixed input conditions
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arXiv
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| Format: | Preprint |
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2024
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| author | Nguyen, Ba Phi Kim, Kihong |
| author_facet | Nguyen, Ba Phi Kim, Kihong |
| contents | 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 $\langle T \rangle \propto L^{-γ_a}$ and $\langle \ln T \rangle \approx -γ_g \ln L$ 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 $γ_a$ and $γ_g$ 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 $\ln T$ 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 $β$. As $V$ increases, the region exhibiting power-law localization diminishes and eventually disappears when $V$ exceeds $β$, leading to standard Anderson localization. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09339 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Power-law localization in one-dimensional systems with nonlinear disorder under fixed input conditions Nguyen, Ba Phi Kim, Kihong Disordered Systems and Neural Networks 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 $\langle T \rangle \propto L^{-γ_a}$ and $\langle \ln T \rangle \approx -γ_g \ln L$ 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 $γ_a$ and $γ_g$ 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 $\ln T$ 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 $β$. As $V$ increases, the region exhibiting power-law localization diminishes and eventually disappears when $V$ exceeds $β$, leading to standard Anderson localization. |
| title | Power-law localization in one-dimensional systems with nonlinear disorder under fixed input conditions |
| topic | Disordered Systems and Neural Networks |
| url | https://arxiv.org/abs/2408.09339 |