2017/05/04 by Jianhua Zeng, Boris A. Malomed
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Geometry #Mathematical physics #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Parameter space #Physics #Quantum mechanics #Space (punctuation) #Vortex #nlin.PS #physics.optics #quant-ph
paper · pdf · doi:10.1103/physreve.95.052214
published as Phys. Rev. E 95, 052214 (2017) · 9 pages,9 figures,Physical Review E, in press
arxiv created 2017/05/04 · openalex publication_date 2017/05/24 · arxiv updated 2017/05/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Recent studies have demonstrated that defocusing cubic nonlinearity with local strength growing from the center to the periphery faster than rD, in space of dimension D with radial coordinate r, supports a vast variety of robust bright solitons. In the framework of the same model, but with a weaker spatial-growth rate ∼rα with α≤D, we test here the possibility to create stable localized continuous waves (LCWs) in one-dimensional (1D) and 2D geometries, localized dark solitons (LDSs) in one dimension, and localized dark vortices (LDVs) in two dimensions, which are all realized as loosely confined states with a divergent norm. Asymptotic tails of the solutions, which determine the divergence of the norm, are constructed in a universal analytical form by means of the Thomas-Fermi approximation (TFA). Global approximations for the LCWs, LDSs, and LDVs are constructed on the basis of interpolations between analytical approximations available far from (TFA) and close to the center. In particular, the interpolations for the 1D LDS, as well as for the 2D LDVs, are based on a deformed-tanh expression, which is suggested by the usual 1D dark-soliton solution. The analytical interpolations produce very accurate results, in comparison with numerical findings, for the 1D and 2D LCWs, 1D LDSs, and 2D LDVs with vorticity S=1. In addition to the 1D fundamental LDSs with the single notch and 2D vortices with S=1, higher-order LDSs with multiple notches are found too, as well as double LDVs, with S=2. Stability regions for the modes under consideration are identified by means of systematic simulations, the LCWs being completely stable in one and two dimensions, as they are ground states in the corresponding settings. Basic evolution scenarios are identified for those vortices that are unstable. The settings considered in this work may be implemented in nonlinear optics and in Bose-Einstein condensates.