TL;DR
This paper introduces an extended lattice Gelfand-Dickey hierarchy with additional logarithmic flows, providing new integrable structures and equations that generalize existing lattice hierarchies.
Contribution
It proposes a novel extension of the lattice Gelfand-Dickey hierarchy incorporating infinite logarithmic flows, with new Lax, Sato, Hirota equations, and a difference operator factorization.
Findings
Formulation of the extended hierarchy with logarithmic flows
Derivation of Lax, Sato, and Hirota equations for the extended system
Establishment of a factorization problem capturing all solutions
Abstract
The lattice Gelfand-Dickey hierarchy is a lattice version of the Gelfand-Dickey hierarchy. A special case is the lattice KdV hierarchy. Inspired by recent work of Buryak and Rossi, we propose an extension of the lattice Gelfand-Dickey hierarchy. The extended system has an infinite number of logarithmic flows alongside the usual flows. We present the Lax, Sato and Hirota equations and a factorization problem of difference operators that captures the whole set of solutions. The construction of this system resembles the extended 1D and bigraded Toda hierarchy, but exhibits several novel features as well.
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