随机势中的非线性格波
Nonlinear Lattice Waves in Random Potentials
摘要：
无序波的局域化是一个基本的物理问题，包括金属-绝缘体转变、量子霍尔效应、光在光子晶体中的传播以及光学阵列中超冷原子的动力学等方面的理论、实验和数值研究。大强度的光可以引起非线性响应，超冷原子气体可以被调谐到相互作用区域，这又导致平均场水平上的非线性波方程。无序和非线性之间的相互作用，它们的局域化和离域效应是目前该领域的一个有趣和具有挑战性的问题。我们将讨论随机势中非线性格波动力学的最新进展。在波动方程中没有非线性项时，安德森局域化导致波包传播的停止。非线性耦合局域本征态，并可能使安德森局域化的扩展和破坏，由于不可积性，混沌和退相干。扩散过程的特点是由于非线性扩散的普遍subdiffusive法律。我们回顾了广泛的计算研究与可调非线性功率的一维和二维系统。我们还简要地讨论了扩展到其他情况下，线性波动方程的特点本地化：Aubry-Andre本地化与准周期电位，Wannier-Stark本地化与直流场，动力学本地化的动量空间踢转子。
英文摘要：
Localization of waves by disorder is a fundamental physical problem encompassing a diverse spectrum of theoretical, experimental and numerical studies in the context of metal-insulator transition, quantum Hall effect, light propagation in photonic crystals, and dynamics of ultra-cold atoms in optical arrays. Large intensity light can induce nonlinear response, ultracold atomic gases can be tuned into an interacting regime, which leads again to nonlinear wave equations on a mean field level. The interplay between disorder and nonlinearity, their localizing and delocalizing effects is currently an intriguing and challenging issue in the field. We will discuss recent advances in the dynamics of nonlinear lattice waves in random potentials. In the absence of nonlinear terms in the wave equations, Anderson localization is leading to a halt of wave packet spreading. Nonlinearity couples localized eigenstates and, potentially, enables spreading and destruction of Anderson localization due to nonintegrability, chaos and decoherence. The spreading process is characterized by universal subdiffusive laws due to nonlinear diffusion. We review extensive computational studies for one- and two-dimensional systems with tunable nonlinearity power. We also briefly discuss extensions to other cases where the linear wave equation features localization: Aubry-Andre localization with quasiperiodic potentials, Wannier-Stark localization with dc fields, and dynamical localization in momentum space with kicked rotors.
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