Peierls-Nabarro energy surfaces and directional mobility of discrete solitons in two-dimensional saturable nonlinear Schr\"odinger lattices

TL;DR

This paper investigates how two-dimensional discrete solitons in saturable nonlinear Schrödinger lattices can move directionally with minimal radiation, using Peierls-Nabarro energy surfaces to predict optimal mobility regimes and effects of anisotropy.

Contribution

It introduces a numerical method to compute Peierls-Nabarro energy surfaces in 2D lattices, linking surface features to soliton mobility and anisotropy effects.

Findings

Flatter energy surface regions enable better soliton mobility.

Transverse oscillations are influenced by surface curvature.

Weak anisotropy alters mobility and surface topology.

Abstract

We address the problem of directional mobility of discrete solitons in two-dimensional rectangular lattices, in the framework of a discrete nonlinear Schr\"odinger model with saturable on-site nonlinearity. A numerical constrained Newton-Raphson method is used to calculate two-dimensional Peierls-Nabarro energy surfaces, which describe a pseudopotential landscape for the slow mobility of coherent localized excitations, corresponding to continuous phase-space trajectories passing close to stationary modes. Investigating the two-parameter space of the model through independent variations of the nonlinearity constant and the power, we show how parameter regimes and directions of good mobility are connected to existence of smooth surfaces connecting the stationary states. In particular, directions where solutions can move with minimum radiation can be predicted from flatter parts of the surfaces. For such mobile solutions, slight perturbations in the transverse direction yield additional transverse oscillations with frequencies determined by the curvature of the energy surfaces, and with amplitudes that for certain velocities may grow rapidly. We also describe how the mobility properties and surface topologies are affected by inclusion of weak lattice anisotropy.