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 on \(\mathbb {R}\). Our interest lies in solutions that arise through bifurcations from the phase-shifted bright soliton of the nonlinear Schrödinger equation. 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 asymptotic orbital 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.