Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2008-03-11
SIGMA 4 (2008), 029, 30 pages
Nonlinear Sciences
Exactly Solvable and Integrable Systems
This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (Ju
Scientific paper
10.3842/SIGMA.2008.029
New reductions for the multicomponent modified Korteveg-de Vries (MMKdV) equations on the symmetric spaces of {\bf 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 $\mathcal{T}_i$, $i=1,2$ 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 $\mathcal{T}_i$ are studied. We illustrate our results by the MMKdV equations related to the algebra $\mathfrak{g}\simeq so(8)$ and derive several new MMKdV-type equations using group of reductions isomorphic to ${\mathbb Z}_{2}$, ${\mathbb Z}_{3}$, ${\mathbb Z}_{4}$.
Gerdjikov Vladimir S.
Kostov Nikolay A.
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