We examine the evolution of a time-varying perturbation signal pumped into a mono-mode fiber in the anomalous dispersion regime. We analytically establish that the perturbation evolves into a conservative pattern of periodic pulses which structures a
nd profiles share close similarity with the so-called soliton-crystal states recently observed in fiber media [see e.g. A. Haboucha et al., Phys. Rev. Atextbf{78}, 043806 (2008); D. Y. Tang et al., Phys. Rev. Lett. textbf{101}, 153904 (2008); F. Amrani et al., Opt. Express textbf{19}, 13134 (2011)]. We derive mathematically and generate numerically a crystal of solitons using time division multiplexing of identical pulses. We suggest that at very fast pumping rates, the pulse signals overlap and create an unstable signal that is modulated by the fiber nonlinearity to become a periodic lattice of pulse solitons which can be described by elliptic functions. We carry out a linear stability analysis of the soliton-crystal structure and establish that the correlation of centers of mass of interacting pulses broadens their internal-mode spectrum, some modes of which are mutually degenerate. While it has long been known that high-intensity periodic pulse trains in optical fibers are generated from the phenomenon of modulational instability of continuous waves, the present study provides evidence that they can also be generated via temporal multiplexing of an infinitely large number of equal-intensity single pulses to give rise to stable elliptic solitons.
The static properties, i.e., existence and stability, as well as the quench-induced dynamics of nonlinear excitations of the vortex-bright type appearing in two-dimensional harmonically confined spin-1 Bose-Einstein condensates are investigated. Line
arly stable vortex-bright-vortex and bright-vortex-bright solutions arise in both antiferromagnetic and ferromagnetic spinor gases upon quadratic Zeeman energy shift variations. The precessional motion of such coherent structures is subsequently monitored dynamically. Deformations of the above configurations across the relevant transitions are exposed and discussed in detail. It is further found that stationary states involving highly quantized vortices can be realized in both settings. Spatial elongations, precessional motion and spiraling of the nonlinear excitations when exposed to finite temperatures and upon crossing the distinct phase boundaries, via quenching of the quadratic Zeeman coefficient, are unveiled. Spin-mixing processes triggered by the quench lead, among others, to changes in the waveform of the ensuing configurations. Our findings reveal an interplay between pattern formation and spin-mixing processes being accessible in contemporary cold atom experiments.
Wave modes induced by cross-phase reshaping of a probe photon in the guiding structure of a periodic train of temporal pulses are investigated theoretically with emphasis on exact solutions to the wave equation for the probe. The study has direct con
nection with recent advances on the issue of light control by light, the focus being on the trapping of a low-power probe by a temporal sequence of periodically matched high-power pulses of a dispersion-managed optical fiber. The problem is formulated in terms of the nonlinear optical fiber equation with averaged dispersion, coupled to a linear equation for the probe including a cross-phase modulation term. Shape-preserving modes which are robust against the dispersion are shown to be induced in the probe, they form a family of mutually orthogonal solitons the characteristic features of which are determined by the competition between the self-phase and cross-phase effects. Considering a specific context of this competition, the theory predicts two degenerate modes representing a train of bright signals and one mode which describes a train of dark signals. When the walk-off between the pump and probe is taken into consideration, these modes have finite-momentum envelopes and none of them is totally transparent vis-`a-vis the optical pump soliton.
The dynamics and stability of continuous-wave and multi-pulse structures are studied theoretically, for a generalized model of passively mode-locked fiber laser with an arbitrary nonlinearity. The model is characterized by a complex Ginzburg-Landau e
quation with saturable nonlinearity of a general form ($I^m/(1+Gamma I)^n$), where $I$ is the field intensity, $m$ and $n$ are two positive real numbers and $Gamma$ is the optical field saturation power. The analysis of fixed-point solutions of the governing equations, reveals an interesting loci of singular points in the amplitude-frequency plane consisting of zero, one or two fixed points depending upon the values of $m$ and $n$. The stability of continuous waves is analyzed within the framework of the modulational-instability theory, results demonstrate a bifurcation in the continuous-wave amplitude growth rate and propagation constant characteristic of multi-periodic wave structures. In the full nonlinear regime these multi-periodic wave structures turn out to be multi-pulse trains, unveiled via numerical simulations of the model nonlinear equation the rich variety of which is highlighted by considering different combinations of values for the pair ($m$,$n$). Results are consistent with previous analyses of the dynamics of multi-pulse structures in several contexts of passively mode-locked lasers with saturable absorber, as well as with predictions about the existence of multi-pulse structures and bound-state solitons in optical fibers with strong optical nonlinearity such as cubic-quintic and saturable nonlinearities.
Growth-induced pattern formations in curved film-substrate structures have attracted extensive attentions recently. In most existing literature, the growth tensor is assumed to be homogeneous or piecewise homogeneous. In this paper, we aim at clarify
ing the influence of a growth gradient on pattern formation and pattern evolution in bilayered tubular tissues under plane-strain deformation. In the framework of finite elasticity, a bifurcation condition is derived for a general material model and a generic growth function. Then we suppose that both layers are composed of neo-Hookean materials. In particular, the growth function is assumed to decay linearly from the inner surface or from the outer surface. It is found that a gradient in the growth has a weak effect on the critical state, compared to the homogeneous growth type where both layers share the same growth factor. Furthermore, a finite element model is built to validate the theoretical model and to investigate the post-buckling behaviors. It is found that the associated pattern transition is not controlled by the growth gradient but by the ratio of the shear modulus between two layers. Different morphologies can occur when the modulus ratio is varied. The current analysis could provide useful insight into the influence of a growth gradient on surface instabilities and suggests that a homogeneous growth field may provide a good approximation on interpreting complicated morphological formations in multiple systems.
Confined active nematics exhibit rich dynamical behavior, including spontaneous flows, periodic defect dynamics, and chaotic `active turbulence. Here, we study these phenomena using the framework of Exact Coherent Structures, which has been successfu
l in characterizing the routes to high Reynolds number turbulence of passive fluids. Exact Coherent Structures are stationary, periodic, quasiperiodic, or traveling wave solutions of the hydrodynamic equations that, together with their invariant manifolds, serve as an organizing template of the dynamics. We compute the dominant Exact Coherent Structures and connecting orbits in a pre-turbulent active nematic channel flow, which enables a fully nonlinear but highly reduced order description in terms of a directed graph. Using this reduced representation, we compute instantaneous perturbations that switch the system between disparate spatiotemporal states occupying distant regions of the infinite dimensional phase space. Our results lay the groundwork for a systematic means of understanding and controlling active nematic flows in the moderate to high activity regime.
Rogue waves are abnormally large waves which appear unexpectedly and have attracted considerable attention, particularly in recent years. The one space, one time (1+1) nonlinear Schrodinger equation is often used to model rogue waves; it is an envelo
pe description of plane waves and admits the so-called Pergerine and Kuznetov-Ma soliton solutions. However, in deep water waves and certain electromagnetic systems where there are two significant transverse dimensions, the 2+1 hyperbolic nonlinear Schrodinger equation is the appropriate wave envelope description. Here we show that these rogue wave solutions suffer from strong transverse instability at long and short frequencies. Moreover, the stability of the Peregrine soliton is found to coincide with that of the background plane wave. These results indicate that, when applicable, transverse dimensions must be taken into account when investigating rogue wave pheneomena.
Neurons are often connected, spatially and temporally, in phenomenal ways that promote wave propagation. Therefore, it is essential to analyze the emergent spatiotemporal patterns to understand the working mechanism of brain activity, especially in c
ortical areas. Here, we present an explicit mathematical analysis, corroborated by numerical results, to identify and investigate the spatiotemporal, non-uniform, patterns that emerge due to instability in an extended homogeneous 2D spatial domain, using the excitable Izhikevich neuron model. We examine diffusive instability and perform bifurcation and fixed-point analyses to characterize the patterns and their stability. Then, we derive analytically the amplitude equations that establish the activities of reaction-diffusion structures. We report on the emergence of diverse spatial structures including hexagonal and mixed-type patterns by providing a systematic mathematical approach, including variations in correlated oscillations, pattern variations and amplitude fluctuations. Our work shows that the emergence of spatiotemporal behavior, commonly found in excitable systems, has the potential to contribute significantly to the study of diffusively-coupled biophysical systems at large.
The paper is devoted to the dynamics of dissipative gap solitons in the periodically corrugated optical waveguides whose spectrum of linear excitations contains a mode that can be referred to as a quasi-Bound State in the Continuum. These systems can
support a large variety of stable bright and dark dissipative solitons that can interact with each other and with the inhomogeneities of the pump. One of the focus points of this work is the influence of slow variations of the pump on the behavior of the solitons. It is shown that for the fixed sets of parameters the effect of pump inhomogeneities on the solitons is not the same for the solitons of different kinds. The second main goal of the paper is systematic studies of the interaction between the solitons of the same or of different kinds. It is demonstrated that various scenarios of inter-soliton interactions can occur: the solitons can repulse each other or get attracted. In the latter case, the solitons can annihilate, fuse in a single soliton or form a new bound state depending on the kinds of the interacting solitons and on the system parameters.
In the present work we illustrate that classical but nonlinear systems may possess features reminiscent of quantum ones, such as memory, upon suitable external perturbation. As our prototypical example, we use the two-dimensional complex Ginzburg-Lan
dau equation in its vortex glass regime. We impose an external drive as a perturbation mimicking a quantum measurement protocol, with a given measurement rate (the rate of repetition of the drive) and mixing rate (characterized by the intensity of the drive). Using a variety of measures, we find that the system may or may not retain its coherence, statistically retrieving its original glass state, depending on the strength and periodicity of the perturbing field. The corresponding parametric regimes and the associated energy cascade mechanisms involving the dynamics of vortex waveforms and domain boundaries are discussed.
