Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings

Boris A. Malomed*

*Corresponding author for this work

Research output: Contribution to journalReview articlepeer-review

1 Scopus citations

Abstract

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross–Pitaevskii (GP) equations with the Lee–Huang–Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole–dipole and quadrupole–quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic ((Formula presented.)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its (Formula presented.) -symmetric version, a system of DNLS equations for the spin–orbit-coupled (SOC) binary Bose–Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and (Formula presented.) -symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

Original languageEnglish
Article number137
JournalEntropy
Volume26
Issue number2
DOIs
StatePublished - Feb 2024
Externally publishedYes

Funding

FundersFunder number
Israel Science Foundation1695/22

    Keywords

    • Bose–Einstein condensates
    • PT symmetry
    • Salerno model
    • lattice dynamics
    • nonlinear Schrödinger equations
    • nonlinear optics
    • quantum droplets
    • skyrmions
    • spin–orbit coupling
    • vortices

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