Solitary waves in a two-dimensional nonlinear Dirac equation: from discrete to continuum

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 50; no. 49; pp. 495207 - 495223
• Main Authors: Cuevas-Maraver, J, Kevrekidis, P G, Aceves, A B, Saxena, Avadh
• Format: Journal Article
• Language: English
• Published: IOP Publishing 08.12.2017
• Subjects: discrete solitons; nonlinear Dirac equation; solitons
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/aa8e36

In the present work, we explore a nonlinear Dirac equation motivated as the continuum limit of a binary waveguide array model. We approach the problem both from a near-continuum perspective as well as from a highly discrete one. Starting from the former, we see that the continuum Dirac solitons can be continued for all values of the discretization (coupling) parameter, down to the uncoupled (so-called anti-continuum) limit where they result in a 9-site configuration. We also consider configurations with 1- or 2-sites at the anti-continuum limit and continue them to large couplings, finding that they also persist. For all the obtained solutions, we examine not only the existence, but also the spectral stability through a linearization analysis and finally consider prototypical examples of the dynamics for a selected number of cases for which the solutions are found to be unstable.