Generalized Volterra lattices: binary Darboux transformations and self-consistent sources

TL;DR

This paper explores matrix generalized Volterra lattices, deriving binary Darboux transformations and self-consistent sources through bidifferential calculus, and constructs exact solutions from basic seed solutions.

Contribution

It introduces new matrix versions of Volterra lattices, applying bidifferential calculus to derive transformations and sources, and constructs explicit solutions.

Findings

Derived binary Darboux transformations for the lattices

Established self-consistent source extensions

Constructed exact solutions from seed solutions

Abstract

We study two families of (matrix versions of) generalized Volterra (or Bogoyavlensky) lattice equations. For each family, the equations arise as reductions of a partial differential-difference equation in one continuous and two discrete variables, which is a realization of a general integrable equation in bidifferential calculus. This allows to derive a binary Darboux transformation and also self-consistent source extensions via general results of bidifferential calculus. Exact solutions are constructed from the simplest seed solutions.