A systematic construction of integrable delay-difference and delay-differential analogues of soliton equations

TL;DR

This paper introduces a systematic method to construct integrable delay-difference and delay-differential equations as analogues of classical soliton equations, preserving multi-soliton solutions and connecting to known equations as delays vanish.

Contribution

It develops a unified approach to derive delay analogues of soliton equations from discrete integrable systems using reduction and delay limits.

Findings

Constructed delay-difference and delay-differential equations with multi-soliton solutions.

Demonstrated convergence of solutions to classical soliton equations as delay parameters approach zero.

Provided a framework linking discrete integrable systems to their delay analogues.

Abstract

We propose a systematic method for constructing integrable delay-difference and delay-differential analogues of known soliton equations such as the Lotka-Volterra, Toda lattice, and sine-Gordon equations and their multi-soliton solutions. It is carried out by applying a reduction and delay-differential limit to the discrete KP or discrete two-dimensional Toda lattice equations. Each of the delay-difference and delay-differential equations has the N-soliton solution, which depends on the delay parameter and converges to an N-soliton solution of a known soliton equation as the delay parameter approaches 0.