Spontaneous patterns from Ablowitz-Ladik equations:
cavity boundary conditions, instabilities, and mean-field theory

Christian, JM; Moorcroft, TT

Abstract

In physics, the discrete nonlinear Schrödinger (dNLS) equation plays a key role in modelling wave
propagation in periodic systems. Optical architectures typically involve light confined to a set of
waveguide channels with nearest-neighbour coupling and whose dielectric response has a local cubic
nonlinearity. While the widely-used dNLS model is non-integrable, it possesses an exactly-integrable
counterpart––the Ablowitz-Ladik (AL) equation––which is often of greater interest in applied
mathematics contexts. The trade-off for introducing integrability is a nonlinearity in the AL equation
that remains cubic but which becomes nonlocal in a way that eludes straightforward physical
interpretations. Despite their subtle differences, both models share common asymptotic properties in
the long-wavelength (continuum) limit.
Here, the pattern-forming properties of the AL equation are explored in detail. Linear analysis in
conjunction with periodic longitudinal boundary conditions––mimicking feedback in an optical
cavity––is deployed. Threshold spectra for static patterns are calculated, and simulations test those
theoretical predictions in AL systems with both one and two transverse dimensions. Subject to
perturbed plane-wave pumping, we find the emergence of stable cosine-type and hexagon patterns,
respectively. We conclude with an excursion into mean-field theory, where the AL equation takes on
the guise of a discrete Ginzburg-Landau model.

Citation

cavity boundary conditions, instabilities, and mean-field theory. Presented at 4th IMA Conference on Nonlinearity and Coherent Structures, University of Loughborough (online)

• Presentation Conference Type: Other
• Conference Name: 4th IMA Conference on Nonlinearity and Coherent Structures
• Conference Location: University of Loughborough (online)
• End Date: Jul 9, 2021
• Publication Date: Jul 7, 2021
• Deposit Date: Jul 12, 2021
• Publicly Available Date: Jul 12, 2021
• Publisher URL: https://cdn.ima.org.uk/wp/wp-content/uploads/2020/10/Abstract-Book-4th-IMA-Conference-on-Nonlinearity-and-Coherent-Structures-2-2.pdf
• Related Public URLs: https://ima.org.uk/15373/4th-ima-conference-on-nonlinearity-and-coherent-structures/
• Additional Information: Event Type : Conference