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

TL;DR

This paper derives exact periodic and pulse-like solutions for the 2D discrete nonlinear Schrödinger equation with saturable nonlinearity, analyzes their stability, and finds the Peierls-Nabarro barrier to be zero for pulse solutions.

Contribution

It provides the first explicit solutions and stability analysis for the 2D discrete nonlinear Schrödinger equation with saturable nonlinearity.

Findings

Existence of exact periodic and pulse solutions

Stability diagrams for these solutions

Zero Peierls-Nabarro barrier for pulse solutions

Abstract

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.