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


SIGMA 4 (2008), 029, 30 pages      arXiv:0803.1651      http://dx.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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