A geometric construction of solutions of the strict $\mathbf h$-hierarchy
G. F. Helminck Korteweg-de Vries Institute, University of Amsterdam, Amsterdam, The Netherlands
Abstract:
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.
Keywords:
matrix pseudodifferential operator, Lax equation, strict $\mathbf{h}$-hierarchy, linearization, matrix wave function. Received: 22.12.2018
Revised: 22.12.2018
DOI:
10.4213/tmf9696
Fulltext:
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