A~geometric construction of solutions of the~strict $\mathbf h$-hierarchy

G. F. Helminck

Teoretičeskaâ i matematičeskaâ fizika, Tome 200 (2019) no. 1, pp. 72-95

Résumé

Let $\mathbf{h}$ be a complex commutative subalgebra of the $n{\times}n$ matrices $M_n(\mathbb{C})$. In the algebra MPsd of matrix pseudodifferential operators in the derivation $\partial$, we previously considered deformations of $\mathbf{h}[\partial]$ and of its Lie subalgebra $\mathbf{h}[\partial]_{>0}$ consisting of elements without a constant term. It turned out that the different evolution equations for the generators of these two deformed Lie algebras are compatible sets of Lax equations and determine the corresponding $\mathbf{h}$-hierarchy and its strict version. Here, with each hierarchy, we associate an $MPsd$-module representing perturbations of a vector related to the trivial solution of each hierarchy. In each module, we describe so-called matrix wave functions, which lead directly to solutions of their Lax equations. We next present a connection between the matrix wave functions of the $\mathbf{h}$-hierarchy and those of its strict version; this connection is used to construct solutions of the latter. The geometric data used to construct the wave functions of the strict $\mathbf{h}$-hierarchy are a plane in the Grassmanian $Gr(H)$, a set of $n$ linearly independent vectors $\{w_i\}$ in $W$, and suitable invertible maps $\delta\colon S^1\to\mathbf{h}$, where $S^1$ is the unit circle in $\mathbb{C}^*$. In particular, we show that the action of a corresponding flow group can be lifted from $W$ to the other data and that this lift leaves the constructed solutions of the strict $\mathbf{h}$-hierarchy invariant. For $n>1$, it can happen that we have different solutions of the strict $\mathbf{h}$-hierarchy for fixed $W$ and $\{w_i\}$. We show that they are related by conjugation with invertible matrix differential operators.

Détail

Keywords: matrix pseudodifferential operator, strict $\mathbf{h}$-hierarchy, linearization, matrix wave function.
Mots-clés : Lax equation