Long-Range Interaction and Synchronization of Oscillating Dissipative Solitons

Abstract

We study the interaction of well-separated oscillating localized structures (oscillons). We show that oscillons emit weakly decaying dispersive waves, which lead to the formation of bound states due to harmonic synchronization. We also show that in optical applications the Andronov-Hopf bifurcation of stationary localized structures leads to a drastic increase in their interaction strength. DOI: 10.1103/PhysRevLett.108.263906 PACS numbers: 42.65.Sf, 47.54.Àr The investigation of localized structures arising in physical systems of various nature is an important subject of nonlinear science. Lately a lot of attention has been paid to the so-called dissipative solitons [1]. Their formation requires a balance of energy gain and dissipation, which makes the dissipative solitons more stable to perturbations and, therefore, more attractive for practical applications (e.g., for optical information processing) than the classical solitons of integrable Hamiltonian equations. Exact analytical expressions for dissipative solitons are rarely available, so qualitative methods become especially important in their study. An interesting problem which can be treated by qualitative methods is the interaction of dissipative solitons [2][3][4][5][6][7][8]. While most of the studies here were focused on the case of stationary solitons, in this Letter we analyze the interaction of dissipative solitons which oscillate in time.It is well known that a stationary soliton can exhibit instabilities that lead to various dynamical regimes. One of the simplest and most frequently encountered between these instabilities is the Andronov-Hopf (AH) bifurcation resulting in undamped pulsations of the soliton's parameters, such as amplitude, width, etc. [1,[8][9][10]. Here, we show that the transition from stationary to an oscillating soliton (oscillon) leads to the formation of various new types of multisoliton bound states. In particular, the AH bifurcation of stationary optical pulses results in a considerable increase of their interaction strength.Although the approach we use is general, to illustrate the enhancement of the solitons' interaction, we consider a specific model equation (Lugiato-Lefever model [11])which describes formation of transverse patterns in Kerr cavity [10] or ''temporal cavity solitons'' in fibers [12]. Here, a is the field envelope, is the cavity decay rate, is the cavity detuning, and p is the external coherent pumping. Spatial filtering (or, in time domain, gain dispersion) is typically quite small in optical applications.The soliton in Eq. (1) is asymptotic to a nonzero stationary value a s ðpÞ. The existence region of the soliton is presented in the bifurcation diagram in Fig. 1. As the pumping parameter p increases above the critical value p AH , the soliton undergoes an AH bifurcation [10]. After this bifurcation, the soliton starts to oscillate, and the oscillations in the tail can be interpreted as a radiation of evanescent waves, see Fig. 2 for an illustration.In order to find the dispersion relation which de...

Dynamics of localized and patterned structures in the Lugiato-Lefever equation determine the stability and shape of optical frequency combs

“…Because of the Kerrinduced tilt of the cavity resonance, the detuning, which is measured from the center of the linear (cold) resonance, is correspondingly large. Only a few recent studies have been reported in this regime [5,19,20]. In particular, a fiber cavity experiment has demonstrated that CSs can display periodic oscillations, and preliminary numerical analysis have also revealed various chaotic regimes [20].…”

Bifurcation structure of cavity soliton dynamics in a vertical-cavity surface-emitting laser with a saturable absorber and time-delayed feedback

“…Previously, oscillatory dynamics of CSs have been observed in systems without optical feedback: in the Lugiato-Lefever model [47] of a driven optical nonlinear cavity [48][49][50] and in a model of a VCSEL with a saturable absorber extended beyond the mean-field approximation [51]. Furthermore, a period doubling route to chaos has been predicted for localized structures in the Lugiato-Lefever equation [52,53] in a forced and damped van der Pol model [54].…”

Periodic and chaotic solitons in a semiconductor laser with saturable absorber

“…Phase-locking is robust against noise, that was always included in our simulations. This phenomenon could be related to the enhancement of solitons' interaction owing to the Andronov-Hopf bifurcation that makes them oscillate, which was demonstrated for Kerr CSs in [35].…”