Reductions of Multicomponent mKdV Equations on Symmetric Spaces of DIII-Type

TL;DR

This paper introduces new reductions for multicomponent mKdV equations on DIII-type symmetric spaces, analyzing their inverse scattering problems, Hamiltonian structures, and deriving novel equations using symmetry groups.

Contribution

It develops new reductions for multicomponent mKdV equations on DIII symmetric spaces and explores their impact on inverse scattering and Hamiltonian hierarchies.

Findings

Defined minimal scattering data sets for unique reconstruction

Analyzed effects of reductions on Hamiltonian structures

Derived new MMKdV-type equations using symmetry groups

Abstract

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 ${\mathbb Z}_{2}$, ${\mathbb Z}_{3}$, ${\mathbb Z}_{4}$.