Dynamics of Localized Structures in Systems with Broken Parity Symmetry

A great variety of nonlinear dissipative systems are known to host structures having a correlation range much shorter than the size of the system. The dynamics of these localized structures (LSs) has been investigated so far in situations featuring parity symmetry. In this Letter we extend this analysis to systems lacking this property. We show that the LS drifting speed in a parameter varying landscape is not simply proportional to the parameter gradient, as found in parity preserving situations. The symmetry breaking implies a new contribution to the velocity field which is a function of the parameter value, thus leading to a new paradigm for LSs manipulation. We illustrate this general concept by studying the trajectories of the LSs found in a passively mode-locked laser operated in the localization regime. Moreover, the lack of parity affects significantly LSs interactions which are governed by asymmetrical repulsive forces.

Figure 1

(a) Profiles of the field intensity ${|E|}^2$, the carrier $G$ and absorption $Q$. (b),(c) Bifurcation diagram of the stable solution branch for the drift velocity and the energy. Parameters are $(\gamma ,\kappa ,\alpha ,\beta ,\mathrm{\Gamma },Q_0,s)=(40,0.8,1,0.5,0.04,0.3,30)$.

Figure 2

(a)–(c) Spatiotemporal diagrams of the LS position evolution (black line) when the current is sinusoidally modulated (color scale in mA) with an amplitude $\delta J=120\mathrm{mA}$ around $J_{cw}=226\mathrm{mA}$ and ${\nu }_m=66614250\mathrm{Hz}$. The detunings are (a) $\mathrm{\Delta }=-4.75\mathrm{kHz}$, (b) $\mathrm{\Delta }=-0.25\mathrm{kHz}$, and (c) $\mathrm{\Delta }=1.75\mathrm{kHz}$. (d) LS drifting speed induced by the variation of $J$ (circle) and best fit with a square-root function.

Figure 3

(a),(b) Spatiotemporal diagrams of the LS position evolution (black line) over the current modulation (color scale in mA). (a) $\mathrm{\Delta }=-9.25\mathrm{k}\mathrm{Hz}$, (b) $\mathrm{\Delta }=3.25\mathrm{k}\mathrm{Hz}$, all other parameters as in Fig. 2. (c) LS drifting speed (red) calculated from the derivative of the trajectory shown in (a) and calculated as ${\upsilon }_{\mathrm{\Delta }}+{\upsilon }_J$ (dots) with ${\upsilon }_J$ obtained fitting the curve ${\upsilon }_J$ in Fig. 2 with $J$ the value of the current along the trajectory.

Figure 4

Experimental spatiotemporal diagrams showing the evolution of three LSs when an electrical square pulse of 4 ns is applied (a) and removed (b) to the pumping current (color scale in mA). The current pulse amplitude is $120\mathrm{m}\mathrm{A}$ and $J_{cw}=220$ mA. In (b) $\mathrm{\Delta }=0$, while $\mathrm{\Delta }$ is slightly negative in (a) explaining the negative drifting speed from $N=0$ until $N=800$. Panel (c): Numerically calculated trajectory using Eqs. (8),(9), with parameters $({\upsilon }_0,\mathrm{\Delta }\upsilon ,P_0,\mathrm{\Delta }P)=(0,-0.01,0.28,0.7)$, $G_{\mathrm{sn}}=0.845G_{\mathrm{th}}$ and ${\upsilon }_{\mathrm{\Delta }}=-2.35\times {10}^{-3}$. Other parameters as in Fig. 1.