L. Bengel, D. Pelinovsky, and W. Reichel
Pinning in the extended Lugiato-Lefever equation
Abstract:
We consider a variant of the Lugiato-Lefever equation (LLE), which is a nonlinear
Schrodinger equation on a one-dimensional torus with forcing and damping, to which
we add a first-order derivative term with a potential. The potential breaks the translation
invariance of LLE. Depending on the existence of zeroes of the effective potential,
which is a suitably weighted and integrated version of the original potential, we show that stationary solutions
without the potential can be continued locally into the range of the nonzero potential. Moreover, the extremal points
of the continued solutions are located near zeros of the effective potential. We therefore call this phenomenon
pinning of stationary solutions. If we assume additionally that the starting stationary
solution is spectrally stable with the simple zero eigenvalue due to translation invariance
being the only eigenvalue on the imaginary axis, we can prove asymptotic stability
or instability of its continuation depending on the sign of the derivative of the effective potential
at its simple zero. The variant of the LLE arises in the description of optical frequency combs in a
Kerr nonlinear ring-shaped microresonator which is pumped by two different continuous
monochromatic light sources of different frequencies and different powers. Our analytical
findings are illustrated by numerical simulations.
Keywords:
Lugiato-Lefever equation, effective potential, pinning of stationary solutions, spectral stability,
asymptotic stability.