Bifurcation structure of periodic patterns in the Lugiato-Lefever equation with anomalous dispersion

Abstract

We study the stability and bifurcation structure of spatially extended patterns arising in nonlinear optical resonators with a Kerr-type nonlinearity and anomalous group velocity dispersion, as described by the Lugiato-Lefever equation. While there exists a one-parameter family of patterns with different wavelengths, we focus our attention on the pattern with critical wave number kc arising from the modulational instability of the homogeneous state. We find that the branch of solutions associated with this pattern connects to a branch of patterns with wave number 2kc. This next branch also connects to a branch of patterns with double wave number, this time 4kc, and this process repeats through a series of 2:1 spatial resonances. For values of the detuning parameter approaching θ = 2 from below the critical wave number kc approaches zero and this bifurcation structure is related to the foliated snaking bifurcation structure organizing spatially localized bright solitons. Secondary bifurcations that these patterns undergo and the resulting temporal dynamics are also studied.

Parametric localized patterns and breathers in dispersive quadratic cavities

“…unstably) and stabilizes in the saddle-node SN P r . After that the periodic state remains stable until SN P l , where it changes stability once more and eventually connects back to A − in a spatial resonance [36]. Within the range of parameters studied in this work the primary pattern always arises subcritically.…”

“…We need to point out that together with the primary pattern arising from the MI, there are many others that emerge all along A + and that connect back to A − . A similar scheme has been described in detail in the context of Kerr cavities [36]. The orange line Fig.…”

Parametric localized patterns and breathers in dispersive quadratic cavities

“…For our particular choice of parameters, the pattern arises subcritically from the MI at S c (i.e., unstably) and stabilizes in the saddle node SN P r . After that the periodic state remains stable until SN P l , where it changes stability once more and eventually connects back to A − in a spatial resonance [36]. Within the range of parameters studied in this paper, the primary pattern always arises subcritically.…”

“…2(a) shows one of such type of secondary patterns. A similar scheme has been described in detail in the context of Kerr cavities [36].…”

Bright-soliton frequency combs and dressed states in chi(2) microresonators

“…( 55) explicitly. In fact, ν plays a role of the Bloch momentum which is now quantised, unlike the one that varies continuously in the theory of the unbound crystal lattices [47] and resonators [48].…”