Self-trapping transition in nonlinear cubic lattices

TL;DR

This paper investigates the critical nonlinearity threshold for energy localization in cubic lattices, introducing a new criterion and method to analyze the transition from delocalized to localized states across different dimensions.

Contribution

It develops a general criterion and a computational method for identifying the localization transition in nonlinear cubic lattices, validated across various lattice types and dimensions.

Findings

Derived a simple criterion for localization transition

Introduced a method to compute dynamically excited frequencies

Validated the criterion with applications to binary lattices

Abstract

We explore the fundamental question of the critical nonlinearity value needed to dynamically localize energy in discrete nonlinear cubic (Kerr) lattices. We focus on the effective frequency and participation ratio of the profile to determine the transition into localization in one-, two-, and three-dimensional lattices. A simple and general criterion is developed, for the case of an initially localized excitation, to define the transition region in parameter space ("dynamical tongue") from a delocalized to a localized profile. We introduce a method for computing the dynamically excited frequencies, which helps us validate our stationary ansatz approach and the effective frequency concept. A general analytical estimate of the critical nonlinearity is obtained, with an extra parameter to be determined.We find this parameter to be almost constant for two-dimensional systems and prove its validity by applying it successfully to two-dimensional binary lattices.