Negative Volterra flows

Abstract

Taking the standard zero curvature approach we derive an infinite set of integrable equations, which taken together form the negative Volterra hierarchy. The resulting equations turn out to be nonlocal, which is usual for the negative flows. However, in some cases the nonlocality can be eliminated. Studying the combined action of both positive (classical) and negative Volterra flows, i.e. considering the differential consequences of equations of the extended Volterra hierarchy, we deduce local equations which seem to be promising from the viewpoint of applications. The presented results give answers to some questions related to the classification of integrable differential-difference equations. We also obtain dark solitons of the negative Volterra hierarchy using an elementary approach.

17 pages, no figures

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Cited by in corpus (4)

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• Two New Integrable Lattice Hierarchies Associated With A Discrete Schrodinger Nonisospectral Problem and Their Infinitely Many Conservation Laws
• Negative flows and non-autonomous reductions of the Volterra lattice
• Solitons of the vector KdV and Yamilov lattices