Stability of solitary wave solutions in the Lugiato-Lefever equation

We analyze the spectral and dynamical stability of solitary wave solutions to the Lugiato-Lefever equation (LLE) on $\mathbb{R}$. Our interest lies in solutions that arise through bifurcations from the phase-shifted bright soliton of the nonlinear Schr\"odinger equation (NLS). These solutions are highly nonlinear, localized, far-from-equilibrium waves, and are the physical relevant solutions to model Kerr frequency combs. We show that bifurcating solitary waves are spectrally stable when the phase angle satisfies $\theta \in (0,\pi)$, while unstable waves are found for angles $\theta \in (\pi,2\pi)$. Furthermore, we establish orbital asymptotical stability of spectrally stable solitary waves against localized perturbations. Our analysis exploits the Lyapunov-Schmidt reduction method, the instability index count developed for linear Hamiltonian systems, and resolvent estimates.