论文标题
递归运算符和MKDV方程的层次结构与$ d_4^{（1）} $，$ d_4^{（2）} $和$ d_4^{（3）} $ kac-moody代数
Recursion operators and the hierarchies of MKdV equations related to $D_4^{(1)}$, $D_4^{(2)}$ and $D_4^{(3)}$ Kac-Moody algebras
论文作者
论文摘要
我们在代数$ d_4 \ simeq so（8）$中构建了三个非级别等级。第一个是使用Coxeter自动形态获得的标准$ C_1 = S_ {α_2} S_ {α_1} S_ {α_3} S_ {α_3} S_ {α_4} $，使用其二脑实现。在第二个中，我们使用$ c_2 = c_1r $，其中$ r $是镜像自动形态。第三个是$ c_3 = s_ {α_2} s_ {α_1} t $，其中$ t $是第3订单的外部自动形态学。对于每个等级，我们在相应的线性子空间$ \ mathfrak $ \ mathfrak {g}^{（k）} $中构建了基础。 MKDV层次结构。我们发现了每个层次结构的紧凑型表达式。最后，我们明确地写了第一个非平凡的MKDV方程式及其哈密顿人。对于$ d_4^{（1）} $，这些实际上是两个MKDV系统，因为在这种情况下，指数$ 3 $具有乘数为2。这些MKDV系统中的每一个都由相对于$ \ partial_x $组成的4个方程。对于$ d_4^{（2）} $，这是相对于$ \ partial_x $的三个方程式的系统。最后，对于$ d_4^{（3）} $，这是一个相对于$ \ partial_x $的两个方程式的系统。
We constructed the three nonequivalent gradings in the algebra $D_4 \simeq so(8)$. The first one is the standard one obtained with the Coxeter automorphism $C_1=S_{α_2} S_{α_1}S_{α_3}S_{α_4}$ using its dihedral realization. In the second one we use $C_2 = C_1R$ where $R$ is the mirror automorphism. The third one is $C_3 = S_{α_2}S_{α_1}T$ where $T$ is the external automorphism of order 3. For each of these gradings we constructed the basis in the corresponding linear subspaces $\mathfrak{g}^{(k)}$, the orbits of the Coxeter automorphisms and the related Lax pairs generating the corresponding mKdV hierarchies. We found compact expressions for each of the hierarchies in terms of the recursion operators. At the end we wrote explicitly the first nontrivial mKdV equations and their Hamiltonians. For $D_4^{(1)}$ these are in fact two mKdV systems, due to the fact that in this case the exponent $3$ has multiplicity 2. Each of these mKdV systems consist of 4 equations of third order with respect to $\partial_x$. For $D_4^{(2)}$ this is a system of three equations of third order with respect to $\partial_x$. Finally, for $D_4^{(3)}$ this is a system of two equations of fifth order with respect to $\partial_x$.
