Spatial solitons under competing linear and nonlinear diffractions

TL;DR

This paper introduces a generalized one-dimensional nonlinear Schrödinger model with competing linear and nonlinear diffraction terms, analyzing spatial solitons' stability, interactions, and collapse phenomena in nonlinear photonic crystals.

Contribution

It develops a comprehensive model incorporating nonlinear diffraction effects, providing analytical, variational, and numerical insights into soliton behavior and stability in optical media.

Findings

Exact stability border for solitons derived analytically

Collapse occurs beyond a critical power threshold

In-phase solitons can merge into pulsons or collapse

Abstract

We introduce a general model which augments the one-dimensional nonlinear Schr\"{o}dinger (NLS) equation by nonlinear-diffraction terms competing with the linear diffraction. The new terms contain two irreducible parameters and admit a Hamiltonian representation in a form natural for optical media. The equation serves as a model for spatial solitons near the supercollimation point in nonlinear photonic crystals. In the framework of this model, a detailed analysis of the fundamental solitary waves is reported, including the variational approximation (VA), exact analytical results, and systematic numerical computations. The Vakhitov-Kolokolov (VK) criterion is used to precisely predict the stability border for the solitons, which is found in an exact analytical form, along with the largest total power (norm) that the waves may possess. Past a critical point, collapse effects are observed, caused by suitable perturbations. Interactions between two identical parallel solitary beams are explored by dint of direct numerical simulations. It is found that in-phase solitons merge into robust or collapsing pulsons, depending on the strength of the nonlinear diffraction.