Conservative Time-Delayed Systems: Theory and Applications

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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.