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Reductions of the strict KP hierarchy
G. F. Helmincka,
E. A. Panasenkob a Korteweg-de Vries Institute, University of Amsterdam, Amsterdam, The Netherlands
b Derzhavin Tambov State University, Tambov, Russia
Abstract:
Let
$R$ be a commutative complex algebra and
$\partial$ be a
$\mathbb{C}$-linear derivation of
$R$ such that all powers of
$\partial$ are
$R$-linearly independent. Let
$R[\partial]$ be the algebra of differential operators in
$\partial$ with coefficients in
$R$ and
$Psd$ be its extension by the pseudodifferential operators in
$\partial$ with coefficients in
$R$. In the algebra
$R[\partial]$, we seek monic differential operators
$\mathbf{M}_n$ of order
$n\ge2$ without a constant term satisfying a system of Lax equations determined by the decomposition of
$Psd$ into a direct sum of two Lie algebras that lies at the basis of the strict KP hierarchy. Because this set of Lax equations is an analogue for this decomposition of the
$n$-KdV hierarchy, we call it the strict
$n$-KdV hierarchy. The system has a minimal realization, which allows showing that it has homogeneity properties. Moreover, we show that the system is compatible, i.e., the strict differential parts of the powers of
$M=(\mathbf{M}_n)^{1/n}$ satisfy zero-curvature conditions, which suffice for obtaining the Lax equations for
$\mathbf{M}_n$ and, in particular, for proving that the
$n$th root
$M$ of
$\mathbf{M}_n$ is a solution of the strict KP theory if and only if
$\mathbf{M}_n$ is a solution of the strict
$n$-KdV hierarchy. We characterize the place of solutions of the strict
$n$-KdV hierarchy among previously known solutions of the strict KP hierarchy.
Keywords:
strict KP hierarchy, reduction, minimal realization, scaling transformation. Received: 02.04.2020
Revised: 02.04.2020
DOI:
10.4213/tmf9916
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