Bose-Einstein Condensates and Multi-Component NLS Models on Symmetric Spaces of BD.I-Type. Expansions over Squared Solutions [chapter]

V. S. Gerdjikov, D. J. Kaup, N. A. Kostov, T. I. Valchev
2011 Nonlinear Science and Complexity  
A special class of multicomponent NLS equations, generalizing the vector NLS and related to the BD.I-type symmetric are shown to be integrable through the inverse scattering method (ISM). The corresponding fundamental analytic solutions are constructing thus reducing the inverse scattering problem to a Riemann-Hilbert problem. We introduce the minimal sets of scattering data T which determines uniquely the scattering matrix and the potential Q of the Lax operator. The elements of T can be
more » ... s of T can be viewed as the expansion coefficients of Q over the 'squared solutions' that are natural generalizations of the standard exponentials. Thus we demonstrate that the the mapping T → Q is a generalized Fourier transform. Special attention is paid to two special representatives of this MNLS with three-component and five components which describe spinor (F = 1 and F = 2, respectively) Bose-Einstein condensates. Theorem 1. The NLEE 28 are equivalent to: i) the equations (19) and ii) the following evolution equations for the generalized Gauss factors of T (λ):
doi:10.1007/978-90-481-9884-9_23 fatcat:c6m4sbjlivf7rellb2gus3skpi