Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 096, 48 pages      arXiv:1108.3990      https://doi.org/10.3842/SIGMA.2011.096
Contribution to the Proceedings of the Conference “Symmetries and Integrability of Difference Equations (SIDE-9)”

Polynomial Bundles and Generalised Fourier Transforms for Integrable Equations on A.III-type Symmetric Spaces

Vladimir S. Gerdjikov a, Georgi G. Grahovski a, b, Alexander V. Mikhailov c and Tihomir I. Valchev a
a) Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chausee, Sofia 1784, Bulgaria
b) School of Mathematical Sciences, Dublin Institute of Technology, Kevin Street, Dublin 8, Ireland
c) Applied Math. Department, University of Leeds, Woodhouse Lane, Leeds, LS2 9JT, UK

Received May 26, 2011, in final form October 04, 2011; Published online October 20, 2011

Abstract
A special class of integrable nonlinear differential equations related to A.III-type symmetric spaces and having additional reductions are analyzed via the inverse scattering method (ISM). Using the dressing method we construct two classes of soliton solutions associated with the Lax operator. Next, by using the Wronskian relations, the mapping between the potential and the minimal sets of scattering data is constructed. Furthermore, completeness relations for the 'squared solutions' (generalized exponentials) are derived. Next, expansions of the potential and its variation are obtained. This demonstrates that the interpretation of the inverse scattering method as a generalized Fourier transform holds true. Finally, the Hamiltonian structures of these generalized multi-component Heisenberg ferromagnetic (MHF) type integrable models on A.III-type symmetric spaces are briefly analyzed.

Key words: reduction group; Riemann-Hilbert problem; spectral decompositions; integrals of motion.

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