Rigorous existence of localized solutions for the driven–lossy saturable discrete Lugiato–Lefever equation

Establish a rigorous existence theory for localized stationary solutions (discrete cavity solitons) of the one-dimensional lattice equation with saturable Kerr nonlinearity and nearest-neighbour coupling, given by δ A_n + (α |A_n|^2 / (1 + |A_n|^2)) A_n + c (A_{n+1} + A_{n-1} - 2 A_n) = P, where δ, α ∈ ℂ may have nonzero imaginary parts modeling linear and nonlinear loss, and P ∈ ℝ is a nonzero pump amplitude. This should extend the existing proofs that only cover the conservative case P = Im(δ) = Im(α) = 0.

Background

The paper studies localized states (discrete cavity solitons) in a discrete optical cavity model, a lattice version of the Lugiato–Lefever equation with saturable nonlinearity. In the stationary regime, the model is δ A_n + (α |A_n|^2 / (1 + |A_n|^2)) A_n + c (A_{n+1} + A_{n-1} - 2 A_n) = P, with δ and α complex to include linear and nonlinear losses and P real as a pump. The authors develop a one-active-site approximation and perform numerical continuation and stability analysis to characterize pinning regions, homoclinic snaking, and isolas.

While rigorous existence of localized solutions has been proven for the special conservative case without pump and loss (P = Im(δ) = Im(α) = 0), a general existence theory for the driven–lossy saturable lattice equation used throughout this work is not yet available. The explicit open problem is to extend these existence results to the full dissipative, driven setting, thereby providing a mathematical foundation for the numerically observed localized patterns.