Exact Solutions of the Two-Dimensional Discrete Nonlinear Schr"odinger Equation with Saturable Nonlinearity

We show that the two-dimensional, nonlinear Schr"odinger lattice with a saturable nonlinearity admits periodic and pulse-like exact solutions. We establish the general formalism for the stability considerations of these solutions and give examples of stability diagrams. Finally, we show that the effective Peierls-Nabarro barrier for the pulse-like soliton solution is zero.

💡 Research Summary

The paper investigates the two‑dimensional discrete nonlinear Schrödinger (DNLS) equation with a saturable nonlinearity, a model that captures the dynamics of light in photonic lattices, Bose‑Einstein condensates in optical arrays, and other discrete media where the nonlinear response saturates at high intensities. The authors first formulate the governing equation for the complex field ψ_{m,n}(t) on a square lattice, where the nonlinear term takes the form |ψ|²/(1+|ψ|²). This saturable term reduces to the usual cubic nonlinearity for small amplitudes but approaches a constant for large amplitudes, thereby limiting the strength of the nonlinearity.

To obtain exact analytical solutions, the authors consider two families of ansätze. The first family consists of plane‑wave (periodic) solutions ψ_{m,n}=A exp