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Register a new userDark solitons in the Lugiato-Lefever equation with normal dispersion
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The regions of existence and stability of dark solitons in the Lugiato-Lefever model with normal chromatic dispersion are described. These localized states are shown to be organized in a bifurcation structure known as collapsed snaking implying the presence of a region in parameter space with a finite multiplicity of dark solitons. For some parameter values dynamical instabilities are responsible for the appearance of oscillations and temporal chaos. The importance of the results for understanding frequency comb generation in microresonators is emphasized.
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The damped driven nonlinear Schrodinger equation (NLSE) has been used to understand a range of physical phenomena in diverse systems. Studying this equation in the context of optical hyper-parametric oscillators in anomalous-dispersion dissipative cavities, where NLSE is usually referred to as the Lugiato-Lefever equation (LLE), we are led to a new, reduced nonlinear oscillator model which uncovers the essence of the spontaneous creation of sharply peaked pulses in optical resonators. We identify attracting solutions for this model which correspond to stable cavity solitons and Turing patterns, and study their degree of stability. The reduced model embodies the fundamental connection between mode synchronization and spatiotemporal pattern formation, and represents a novel class of self-synchronization processes in which coupling between nonlinear oscillators is governed by energy and momentum conservation.
We present a stability analysis of the Lugiato-Lefever model for Kerr optical frequency combs in whispering gallery mode resonators pumped in the anomalous dispersion regime. This article is the second part of a research work whose first part was devoted to the regime of normal dispersion, and was presented in ref. cite{Part_I}. The case of anomalous dispersion is indeed the most interesting from the theoretical point of view, because of the considerable variety of dynamical behaviors that can be observed. From a technological point of view, it is also the most relevant because it corresponds to the regime where Kerr combs are predominantly generated, studied, and used for different applications. In this article, we analyze the connection between the spatial patterns and the bifurcation structure of the eigenvalues associated to the various equilibria of the system. The bifurcation map evidences a considerable richness from a dynamical standpoint. We study in detail the emergence of super- and sub-critical Turing patterns in the system. We determine the areas were bright isolated cavity solitons emerge, and we show that soliton molecules can emerge as well. Very complex temporal patterns can actually be observed in the system, where solitons (or soliton complexes) co-exist with or without mutual interactions. Our investigations also unveil the mechanism leading to the phenomenon of breathing solitons. Two routes to chaos in the system are identified, namely a route via the so called secondary combs, and another via soliton breathers. The Kerr combs corresponding to all these temporal patterns are analyzed in detail, and a discussion is led about the possibility to gain synthetic comprehension of the observed spectra out of the dynamical complexity of the system.
We raise a detuning-dependent loss mechanism to describe the soliton formation dynamics when the lumped filtering operation is manipulated in anomalous group velocity dispersion regime, using stability analysis of generalized Lugiato-Lefever equation.
We analyze dark pulse Kerr frequency combs in optical resonators with normal group-velocity dispersion using the Lugiato-Lefever model. We show that in the time domain these correspond to interlocked switching waves between the upper and lower homogeneous states, and explain how this fact accounts for many of their experimentally observed properties. Modulational instability does not play any role in their existence. Furthermore, we provide a detailed map indicating where stable dark pulse Kerr combs can be found in parameter space, and how they are destabilized for increasing values of frequency detuning.
We present the study of the dark soliton dynamics in an inhomogenous fiber by means of a variable coefficient modified nonlinear Schr{o}dinger equation (Vc-MNLSE) with distributed dispersion, self-phase modulation, self-steepening and linear gain/loss. The ultrashort dark soliton pulse evolution and interaction is studied by using the Hirota bilinear (HB) method. In particular, we give much insight into the effect of self-steepening (SS) on the dark soliton dynamics. The study reveals a shock wave formation, as a major effect of SS. Numerically, we study the dark soliton propagation in the continuous wave background, and the stability of the soliton solution is tested in the presence of photon noise. The elastic collision behaviors of the dark solitons are discussed by the asymptotic analysis. On the other hand, considering the nonlinear tunneling of dark soliton through barrier/well, we find that the tunneling of the dark soliton depends on the height of the barrier and the amplitude of the soliton. The intensity of the tunneling soliton either forms a peak or valley and retains its shape after the tunneling. For the case of exponential background, the soliton tends to compress after tunneling through the barrier/well.
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