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

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.