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

TL;DR

This paper analyzes the bifurcation structure of periodic patterns in the Lugiato-Lefever equation with anomalous dispersion, revealing a series of resonances and connections to localized solitons in nonlinear optical resonators.

Contribution

It uncovers the detailed bifurcation and resonance structure of patterns in the Lugiato-Lefever equation, linking spatial patterns to localized solitons and secondary bifurcations.

Findings

Patterns connect through 2:1 spatial resonances.

Bifurcation structure relates to foliated snaking of solitons.

Secondary bifurcations lead to complex temporal dynamics.

Abstract

We study the stability and bifurcation structure of spatially extended patterns arising in nonlin- ear 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 k c 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 $2k_c$ . This next branch also connects to a branch of patterns with double wave number, this time $4k_c$ , and this process repeats through a series of 2:1 spatial resonances. For values of the detuning parameter approaching $\theta = 2$ from below the critical wave number $k_c$ 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.