Construction of the Lax pairs for the delay Lotka-Volterra and delay Toda lattice equations and their reductions to delay Painleve equations

Abstract: The delay Lotka-Volterra and delay Toda lattice equations are delay-differential extensions of the well-known soliton equations, the Lotka-Volterra and Toda lattice equations, respectively. This paper investigates integrable properties of the delay Lotka-Volterra and delay Toda lattice equations, and study the relationships to the already known delay Painleve equations. First, Backlund transformations, Lax pairs and an infinite number of conserved quantities of these delay soliton equations are constructed. Then, applying spatial 2-periodic reductions to them, we show the known delay Painleve equations are derived. Using these reductions, we construct the N-soliton-type determinant solutions of the autonomous versions of delay Painleve equations, and the Casorati determinant solution of a higher order analogue of the discrete Painleve II equation.