This paper studies how light can self-focus and stay localized at the edge of a two‑dimensional waveguide lattice when linear gain and loss are present. The authors work with “dissipative solitons,” which are self‑localized light beams sustained not just by the material’s nonlinearity but also by a continuous balance between local gain and loss. Their focus is on “surface” solitons that sit at a truncated boundary of the lattice, where the geometry of the cut changes what modes are possible.

The research uses a standard wave‑propagation model: a nonlinear Schrödinger equation with a complex potential Vr − iVi. Vr describes the refractive‑index lattice and Vi describes spatially distributed linear gain or loss. The nonlinearity is Kerr type (intensity‑dependent refractive index), with σ = 1 for focusing and σ = −1 for defocusing. For most examples the authors fix lattice parameters pr = 4 (lattice depth), pi = 1 (gain/loss amplitude scale), period d = 4.5, and waveguide width w = 1.5, and then vary the number of waveguide rows kept at the interface and the gain parameter γ.

They first analyze the linear spectrum of truncated lattices to find surface‑localized linear modes. When nonlinearity is turned on, families of dissipative surface solitons (DSSs) emerge from those linear gain‑localized modes inside spectral gaps. Increasing the number of retained transverse rows (single, double, triple rows) produces more distinct surface mode families. The authors find a critical value of the gain parameter near γ0 ≈ −0.657 where the imaginary part of the linear surface eigenvalue changes sign, marking the switch between modes that act like net loss or net gain in the linear regime.

To find nonlinear stationary states the team uses a Newton iteration constrained by a global power‑balance condition, ensuring the total gain and loss cancel for a steady state. They test stability with a linear perturbation (eigenvalue) analysis and with direct propagation simulations. A central result is that dynamical stability is strongly phase selective: in‑phase surface solitons (neighboring peaks having the same phase) can be stable, while out‑of‑phase and antisymmetric surface states become unstable under propagation.