联想记忆复Ginzburg-Landau方程的孤子不动点吸引子
Solitonic fixed point attractors in the complex Ginzburg-Landau equation for associative memories
摘要：
最近显示[V. V. Cherny，T. Byrnes，A.N. Pyrkov，\textit{Adv. Quantum Technol.} \textbf{2019} \textit{2}，1800087]，具有特殊类型的简化耗散扰动的非线性薛定谔方程的特征在于非零振幅的零速度孤子解，其可以用于类比Hopfield关联记忆的吸引子。在这项工作中，我们考虑了一个更复杂的耗散微扰添加双光子吸收的影响和五次增益/损失的影响，正式产生复杂的金兹伯格-朗道方程（CGLE）。我们构造了一个微扰理论的CGLE与一个小的耗散扰动和定义的孤子解的行为与系统的参数和比较的解决方案与数值模拟的CGLE。我们发现，类似于非线性薛定谔方程与简化的耗散项，零速度孤子解的非零振幅出现作为吸引子的CGLE。在这种情况下，孤子固定点吸引子的振幅和速度不依赖于五次增益/损耗效应。此外，双光子吸收效应导致孤子不动点吸引子的强度增加。
英文摘要：
It was recently shown [V.V. Cherny, T. Byrnes, A.N. Pyrkov, \textit{Adv. Quantum Technol.} \textbf{2019} \textit{2}, 1800087] that the nonlinear Schrodinger equation with a simplified dissipative perturbation of special kind features a zero-velocity solitonic solution of non-zero amplitude which can be used in analogy to attractors of Hopfield's associative memory. In this work, we consider a more complex dissipative perturbation adding the effect of two-photon absorption and the quintic gain/loss effects that yields formally the complex Ginzburg-Landau equation (CGLE). We construct a perturbation theory for the CGLE with a small dissipative perturbation and define the behavior of the solitonic solutions with parameters of the system and compare the solution with numerical simulations of the CGLE. We show that similarly to the nonlinear Schrodinger equation with a simplified dissipation term, a zero-velocity solitonic solution of non-zero amplitude appears as an attractor for the CGLE. In this case the amplitude and velocity of the solitonic fixed point attractor does not depend on the quintic gain/loss effects. Furthermore, the effect of two-photon absorption leads to an increase in the strength of the solitonic fixed point attractor.
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