New Lax Representation and Integrable Discretization of the Relativistic Volterra Lattice

The Algebro-Geometric Initial Value Problem for the Relativistic Lotka-Volterra Hierarchy and Quasi-Periodic Solutions

“…(2.69) Remark 3. (i) Taking the transformation p → hp, q → hq, one concludes that the equation (2.35) and ( 2.50) change into the normal form of relativistic Lotka-Volterra system [2,10].…”

“…Mathematical structure related to relativistic volterra lattice (1.5) such as Lax integrablity [3], 2 × 2 Lax representation [10], conservation laws [11], bilinear structure and determinant solution [12] have been closely studied and hence the purpose of this paper is to uniformly construct algebro-geometric solutions of the relativistic Lotka-Volterra hierarchy which invariably is connected with geometry and Riemann theta functions, parameterized by some Riemann surface. Algebrogeometric solutions (finite-gap solutions or quasi-period solutions), as an important character of integrable system, is a kind of explicit solutions closely related to the inverse spectral theory [15,17].…”

Infinite number of conservation laws and Darboux transformations for a 6-field integrable lattice system

“…Mathematical structures related to (4) such as Lax pairs, 21 Hamiltonian structure, master symmetries, infinitely many conservation laws, 22 bilinear structure, 23 and Darboux transformation 24 have been closely studied. In this paper, we firstly establish infinitely many conservation laws formulaically for (1) by using the Lax pair.…”

On an integrable system related to the relativistic Toda lattice – Bäcklund transformation and integrable discretization

“…ϕ n,t = V n (u, λ)ϕ n , (1. 13) where u is the potential function, λ the spectral parameter and E the shift operator defined by Ef n = f n+1 . The integrability condition between (1.12) and (1.13) leads to the integrable lattice system…”

“…The Lagrangian form of dRTL was presented in [13]. The Casorati determinant solution for the discrete RT lattice (1.17) was given in [14] and the elliptic solutions in [15].…”