多分量mKdV方程在DIII型对称空间上的约化
Reductions of Multicomponent mKdV Equations on Symmetric Spaces of DIII-Type
摘要：
利用A. V. Mikhailov引入的约化群方法，在{ \bf DIII}型对称空间上导出了多分量修正Korteveg-de Vries（MMKdV）方程的新约化.相关的逆散射问题进行了研究，并减少到一个Riemann-Hilbert问题。定义了散射数据的最小集合$\mathcal{T}_i$、$i=1,2$，它们允许唯一地重建散射矩阵和Lax算子的势。研究了新约化对MMKdV和$\mathcal{T}_i$的哈密顿结构的影响。我们用与代数$\mathfrak{g}\simeq so(8)$相关的MMKdV方程来说明我们的结果，并利用同构于${\mathbb Z}_{2}$，${\mathbb Z}_{3}$，${\mathbb Z}_{4}$的约化群导出了几个新的MMKdV型方程.
英文摘要：
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 ${\mathbb Z}_{2}$, ${\mathbb Z}_{3}$, ${\mathbb Z}_{4}$.
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