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SIGMA 4 (2008), 029, 30 pages arXiv:0803.1651
https://doi.org/10.3842/SIGMA.2008.029
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics
Reductions of Multicomponent mKdV Equations on Symmetric Spaces of DIII-Type
Vladimir S. Gerdjikov and Nikolay A. Kostov
Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences,
72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria
Received December 14, 2007, in final form February 27, 2008; Published online March 11, 2008
Abstract
New reductions for the multicomponent modified Korteweg-de Vries (MMKdV) equations on the symmetric spaces of
DIII-type are derived using the approach based on the reduction
group introduced by A.V. Mikhailov. The relevant inverse
scattering problem is studied and reduced to a Riemann-Hilbert
problem. The minimal sets of scattering data Ti,
i = 1, 2 which allow one to reconstruct uniquely both the
scattering matrix and the potential of the Lax operator are
defined. The effect of the new reductions on the hierarchy of
Hamiltonian structures of MMKdV and on Ti are
studied. We illustrate our results by the MMKdV equations related to
the algebra g @ so(8) and derive several new
MMKdV-type equations using group of reductions isomorphic to
Z2, Z3, Z4.
Key words:
multicomponent modified Korteweg-de Vries (MMKdV) equations; reduction group; Riemann-Hilbert problem; Hamiltonian structures.
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