Article ID Journal Published Year Pages File Type
8256261 Physica D: Nonlinear Phenomena 2018 50 Pages PDF
Abstract
The inverse scattering transform (IST) with non-zero boundary conditions at infinity is developed for an m×m matrix nonlinear Schrödinger-type equation which, in the case m=2, has been proposed as a model to describe hyperfine spin F=1 spinor Bose-Einstein condensates with either repulsive interatomic interactions and anti-ferromagnetic spin-exchange interactions (self-defocusing case), or attractive interatomic interactions and ferromagnetic spin-exchange interactions (self-focusing case). The IST for this system was first presented by Ieda et al. (2007) , using a different approach. In our formulation, both the direct and the inverse problems are posed in terms of a suitable uniformization variable which allows to develop the IST on the standard complex plane, instead of a two-sheeted Riemann surface or the cut plane with discontinuities along the cuts. Analyticity of the scattering eigenfunctions and scattering data, symmetries, properties of the discrete spectrum, and asymptotics are derived. The inverse problem is posed as a Riemann-Hilbert problem for the eigenfunctions, and the reconstruction formula of the potential in terms of eigenfunctions and scattering data is provided. In addition, the general behavior of the soliton solutions is analyzed in detail in the 2 × 2 self-focusing case, including some special solutions not previously discussed in the literature.
Related Topics
Physical Sciences and Engineering Mathematics Applied Mathematics
Authors
, , , ,