Discrete Snaking: Multiple Cavity Solitons in Saturable Media

Abstract

Abstract. This paper continues an investigation into a one-dimensional lattice equation that models the light field in a system comprised of a periodic array of pumped optical cavities with saturable nonlinearity. The additional effects of a spatial gradient of the phase of the pump field are studied, which in the presence of loss terms is shown to break the spatial reversibility of the steady problem. Unlike for continuum systems, small symmetry-breaking is argued to not lead directly to moving solitons, but there remains a pinning region in which there are infinitely many distinct stable stationary solitons of arbitrarily large width. These solitons are no-longer arranged in a homoclinic snaking bifurcation diagrams, but instead break up into discrete isolas. For large enough symmetry-breaking, the fold bifurcations of the lowest intensity solitons no longer overlap, which is argued to be the trigger point of moving localised structures. Due to the dissipative nature of the problem, any radiation shed by these structures is damped and so they appear to be true attractors. Careful direct numerical simulations reveal that branches of the moving solitons undergo unsual hysteresis with respect to the pump, for sufficiently large symmetry breaking. Introduction.Recently, the present authors [33,34] studied the rich variety of stable localised structures that can occur in a spatially discrete model for an optical cavity with an imposed periodic structure and saturable nonlinearity. In particular, it was shown how infinitely many distinct stable localised structures can occur for a wide range of parameter values due to the so-called homoclinic snaking mechanism that has received a lot of attention in continuum models of pattern formation [1,4,5,6,8,14,20,22,35]. This paper explores the consequences of introducing a spatial gradient to the optical pumping field (the forcing term in the model), which breaks the spatial reversibility of the steady system.The model can be written in dimensionless form as the one-dimensional lattice equation for a complex field A n ∈ C, n ∈ Z:

Control of discrete cavity solitons in coupled cavities

“…In previous works, existence of bright and dark DCSs and their stability have been investigated [4,6]. Following these research, our simulations showed more variues shapes of multy-hump DCSs in small coupling parameters [8].…”

“…Properties of DCSs are different from those of conventional CSs. In recent years, the existence of DCSs, that result from the interplay between nonlinearity and diffraction as well as gain, losses and internal feedback of the system have been predicted for cubic [3], quadratic [4] and saturable [5,6] nonlinear waveguide arrays in presence of normal and inclined holding beam and their stability probed by a conventional linear stability analysis. The holding beam determined the parameters of DCSs (energy and phase) and controls the process of creation and annihilation of them as well as their evolution.…”

Snaking and isolas of localised states in bistable discrete lattices

“…In many physical situations it is appropriate to use mathematical models for which the spatial dynamics are modelled discretely, rather than as a continuum. Such lattice models arise in many physical applications: in particular we mention recent work in nonlinear optics [26,27,8] and mathematical modelling of processes in developmental biology [10,20,24]. In many applications the existence of domain-filling (almost) regular solutions exhibiting spatial periodicity is of interest.…”

Snaking bifurcations in a self-excited oscillator chain with cyclic symmetry

“…Individual self-excited oscillators are coupled, as e.g. in the work of Yulin and Champneys in [26], where a one-dimensional periodic array of optical cavities pumped by coherent light were studied. As in our results, the effect of discreteness was studied, showing that the pinning region (the parameter interval where the snaking occurs) gets progressively narrower as the continuum limit is approached.…”