Integrable delay-difference and delay-differential analogues of the KdV, Boussinesq, and KP equations

TL;DR

This paper introduces delay-difference and delay-differential versions of the KdV, Boussinesq, and KP equations, each possessing soliton solutions and reducing to classical equations as the delay parameter diminishes.

Contribution

It presents new integrable delay-differential equations with soliton solutions and clarifies their relationships and reductions to known equations.

Findings

Each equation has an N-soliton solution.

Equations reduce to classical forms as delay approaches zero.

Relationships between the delay equations are explicitly established.

Abstract

Delay-difference and delay-differential analogues of the KdV and Boussinesq (BSQ) equations are presented. Each of them has the N-soliton solution and reduces to an already known soliton equation as the delay parameter approaches 0. In addition, a delay-differential analogue of the KP equation is proposed. We discuss its N-soliton solution and the limit as the delay parameter approaches 0. Finally, the relationship between the delay-differential analogues of the KdV, BSQ, and KP equations is clarified. Namely, reductions of the delay KP equation yield the delay KdV and delay BSQ equations.