Conservative Time-Delayed Systems: Theory and Applications

由于有限的信号传播速度，现实世界中的复杂系统会受到时滞的强烈影响，而时滞动力系统已被证明是建模非线性现象的一个肥沃的框架。然而，在物理学中，它们大多局限于耗散动力学的研究。PI小组最近的一项理论研究显示了一个由非线性、时间可逆、守恒时滞系统建模的光子系统的例子。项目记忆旨在为时滞系统中的保守非线性动力学提供理论描述，弥补与耗散时滞系统的研究结果之间的差距。我们的方法在于利用时滞微分代数方程和相应的中立型时滞微分方程来简化三个具有可积孤立波解的著名保守模型系统的时滞实现：非线性薛定谔方程、Korteweg-de-Vries方程和Sine-Gordon方程。特别地，在大延迟极限下，结合解析法、数值方法和路径延拓方法，项目记忆研究了相应时滞系统的可积性和时间可逆性，并研究了高阶效应和约束的影响。因此，我们设想在时滞系统中不仅可以观察到保守孤立波，还可以观察到时间Hermite-Gauss模和Fermi-Pasta-Ulam-Tsingou复现。此外，将在所有情况下分析弱耗散动力学。最后，理论工作将与实验伙伴小组提供的有趣的新实验一起探索。反之亦然，理论见解将有助于更好地控制期望的性质，并帮助实验组解释和调整相关的参数制度。

Real-world complex systems can be strongly influenced by time-delays due to unavoidable finite signal propagation speeds and time-delayed dynamical systems have proven to be a fertile framework for the modeling of nonlinear phenomena. However, in physics, they mostly have been limited to the study of dissipative dynamics. A recent theoretical study by the PI group has shown an example of a photonic system modeled by a nonlinear, time-reversible, conservative time-delayed system. The project MEMORY aims to provide the theoretical description of conservative nonlinear dynamics in time-delayed systems bridging the gap with the results known for dissipative time-delayed systems. Our approach consists in employing time-delayed differential algebraic equations and corresponding neutral delay differential equations to facilitate the time-delayed realizations of three famous conservative model systems admitting integrable solitary wave solutions: the nonlinear Schrödinger equation, the Korteweg-de-Vries equation as well as the sine-Gordon equation. In particular, using a combination of analytical, numerical and path-continuation methods in the limit of large delays, the project MEMORY investigates the integrability and time-reversibility of the corresponding time-delayed systems and studies the influence of high-order effects as well as that of the confinement. As such, we envision the possibility to observe not only conservative solitary waves but also temporal Hermite-Gauss modes and Fermi-Pasta-Ulam-Tsingou recurrence in time-delayed systems. Furthermore, the weakly-dissipative dynamics will be analyzed in all cases. Finally, the theoretical work will be explored together with the intriguing new experiments provided by experimental partner groups. Vice versa, the theoretical insights will contribute to a better control of the desired properties and help the experimental groups to interpret and adjust the relevant parameter regimes.