Lotka–Volterra lattice system: N-fold Darboux transformation, the corresponding integrable lattice family and bi-Hamiltonian structure

Abstract

An one-fold Darboux transformation for the Lotka–Volterra lattice system is first established using a proper gauge transformation matrix. Then, as a result of the N times one-fold Darboux transformation, the corresponding N-fold Darboux transformation of the Lotka–Volterra lattice system is presented, and two exact solution are obtained by the resulting Darboux transformation. Hereafter, its the corresponding iso-spectral integrable lattice family is derived. Using the trace identity, bi-Hamiltonian structure of the Lotka–Volterra integrable family is established, and its Liouville integrability is proven.