Conservative Solitons and Reversibility in Time-Delayed models for Optical Micro-Cavities

We are interested in the dynamics of dispersive temporal pulses in optical micro-cavities. Our work is set within the framework of time-delayed systems. While time-delayed systems are usually considered devoid of the essential dispersive effects, recently it was showed that controllable dispersion may appear naturally in models involving delayed algebraic equations (DAEs) [1,2,3]. DAEs are widely used to model the dynamics of coupled micro-cavities, e.g. electrically driven VECSELs mode-locked by saturable absorber mirrors [1], [2]. There, DAE constrains appear naturally when the electric fields in different parts of a compound cavity should be connected by boundary condition. Moreover, it was shown that, operated in reflection, the gain and absorber microcavities behave as dispersive Gires-Tournois interferometers (GTI) [3]. In this contribution we discuss the existence of reversible conservative time-delayed systems considering a photonic system consisting of a dispersive GTI microcavity with a nonlinear Kerr medium and coupled to a distant external mirror [4], see Fig. 1(a). The dynamical model for the slowly varying electromagnetic field envelopes in the microcavity
$E$
and the external cavity
$Y$
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