Reductions of the strict KP hierarchy

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 $$ P{\kern-1.5pt}sd $$ 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 $$ P{\kern-1.5pt}sd $$ 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.