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

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.

Integrability and asymptotic behaviour of a differential-difference matrix equation

“…We note in addition that equation (2.3) can be found in [20] (see also references therein). The Miura map (2.2) from (2.1) to (2.3) is a matrix version of (1.4) from (1.3) to (1.5).…”

“…The Lax pair (2.27) readily reduces to the well-known Lax pair for the scalar Volterra equation (1.7). Equations (2.24) and (2.25) are known matrix generalisations of the Volterra equation (1.7), and can be found for example in [20] (see also references therein).…”

Rational, semi-rational solution and self-consistent sources extension of the variable-coefficient extended modified Kadomtsev-Petviashvili equation

“…As the sources change the velocities of solitary waves, soliton equations with self-consistent sources can provide variety of dynamics of physical models. Mathematically, the soliton equations with self consistent sources have been studied via inverse scattering methods [54][55][56][57][58], Darboux transformation methods [59-61], Hirota's bilinear methods and Wronskian technique [62][63][64][65], and deformations of binary Darboux transformations [66,67]. In [68], the authors propose a new algebraic method, called the source generation procedure, to construct and solve the soliton equations with self consistent sources.…”

Differential Calculi on Associative Algebras and Integrable Systems

“…In Section 6, we sketch recent work about a deformation of the binary Darboux transformation in bidifferential calculus, leading to integrable equations with sources [6,34].…”