On Multicomponent mKDV Equations on Symmetric Spaces of DIII - Type and Their Reductions

Abstract

New reductions for the multicomponent modified Korteveg de Vries (MMKdV) equations on the symmetric spaces of DIII-type are derived using the approach based on the reduction group introduced by A. 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 $\mathfrak{g} \simeq \mathfrak {so}(8)$ and derive several new MMKdV-type equations using group of reductions isomorphic to $\mathbb{Z}_{2}, \mathbb{Z}_{3},\mathbb{Z}_{4}$.