Abstract
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 constructed thus reducing the inverse scattering problem to a Riemann-Hilbert problem. We introduce the minimal sets of scattering data \(\mathfrak{T}\) which determines uniquely the scattering matrix and the potential Q of the Lax operator. The elements of \(\mathfrak{T}\) 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 mapping \(\mathfrak{T}\to Q\) 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.
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Acknowledgements
One of us (V.S.G.) thanks the organizers of the Conference for their hospitality and for making his participation possible. This research has been supported in part by the National Science Foundation of the USA via grant DMS-0505566 and by the USA Air Force Office of Scientific Research.
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Gerdjikov, V.S., Kaup, D.J., Kostov, N.A., Valchev, T.I. (2011). Bose-Einstein Condensates and Multi-Component NLS Models on Symmetric Spaces of BD.I-Type. Expansions over Squared Solutions. In: Machado, J., Luo, A., Barbosa, R., Silva, M., Figueiredo, L. (eds) Nonlinear Science and Complexity. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9884-9_23
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DOI: https://doi.org/10.1007/978-90-481-9884-9_23
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