Conservative Solitons and Reversibility in Time Delayed Systems

Time delayed dynamical systems have proven to be a fertile framework for the study of physical phenomena. In natural sciences, their uses have been limited to the study of dissipative dynamics. In this Letter, we demonstrate the existence of nonlinear reversible conservative time delayed systems. We consider the example of a dispersive microcavity containing a Kerr medium coupled to a distant external mirror. At low energies and in the long delay limit, a multiscale analysis shows the equivalence with the nonlinear Schrödinger equation. We unveil some of the symmetries and conserved quantities, as well as bright temporal solitons. While elastic collisions occur for shallow wave packets, we observe the lack of integrability at higher energies. We recover the Lugiato-Lefever equation in the weakly dissipative regime.

Figure 1

(a) A schematic of a microcavity with round-trip time ${\tau }_c$ enclosed by two distributed Bragg mirrors with reflectivities $r_{1,2}$. It is coupled to a long external cavity with round-trip time $\tau$ which is closed by a mirror with reflectivity $r_e$. $E$ is the sum of the forward and backward propagating fields that interfere upon the Kerr medium while $Y$ is the field impinging upon the top mirror. (b) The eigenvalue spectrum Eqs. (3) and (4) for different values of $h$ and $r_e$. For $h<2$ and $r_e<1$ (red dots) the spectrum resembles an inverted parabola. When $h\to 2$ (blue dots) it flattens as ${\gamma }_m\tau =\ln r_e$. For $r_e\to 1$, the spectrum converges to the imaginary axis.

Figure 2

(a) Temporal trace obtained by numerically integrating Eqs. (1) and (2) for $\tau =300$. The period of the solution is $T=301.5$ and the inset presents an enlarged view over five round-trips. The energy of the pulses is $\mu =0.0063$. (b) Comparison between the profile of a single pulse of (a) (blue dots) and the analytical solution of the NLS equation (red circles) on a logarithmic scale. The only discrepancy can be seen at the interval boundaries (see inset).

Figure 3

(a) Numerical simulation showing the reversibility of Eq. (6). An asymmetric initial condition [blue circles in (b),(c)] is propagated 300 steps (blue part). Next, at the black dashed line in (a), the operator $R$ is applied. After 300 steps (red part), the final profile shown in (b),(c) with red dots perfectly matches the initial condition after applying $R$ a second time.

Figure 4

Two-time representation of two colliding solitons of Eqs. (1) and (2) in-phase (a),(b) and antiphase (c),(d) as a function of the number of steps $N$. Panels (a) and (c) show the absolute value of the profile while (b) and (d) show their phase. Each pulse has an energy of $\mu =0.005$ and ${\delta }_{1,2}=-{\delta }_c\pm 0.1$ while $\phi =1.95$. The mean carrier frequency $e^{-i{\delta }_{c}t}$ was factored out for clarity in (b),(d).

Figure 5

(a),(b) Evolution of Eqs. (1) and (2) for different initial conditions. Note that the folding factor for these plots is $\tau$, instead of $T$ as in Fig. (4), leading to the drift of the soliton. The initial conditions are (a) a hyperbolic secant (8) with $\mu =0.005$ and (b) a condition which is far away from a steady state. (c),(d) Evolution of different measures to visualize the conservation properties. In (c) all quantities are conserved unless the pulse reaches the edge of the cavity. Then only $Q_{\theta }$ is conserved. (d) The norm ${\int }_0^{\tau }|E_{\theta }{|}^{2}d\sigma$ is not conserved and $2{\int }_0^{\tau }|Y_{\theta }{|}^{2}d\sigma$ as well as $|E_{\theta }(0){|}^2$ show the same imperfections when a soliton exits the cavity. However, $Q_{\theta }$ is conserved.