Rational solutions to the ABS list: Degenerating approach

TL;DR

This paper constructs rational solutions for integrable lattice equations in the ABS list using bilinear methods, and derives solutions for related equations through degeneration, all expressed in Casoratian form.

Contribution

It introduces a systematic approach to generate rational solutions for multiple equations in the ABS list via degeneration from NQC solutions.

Findings

Rational solutions for Q3$_\

Solutions expressed in Casoratian form

Degeneration relations connect solutions across equations

Abstract

In the paper we first construct rational solutions for the Nijhoff-Quispel-Capel (NQC) equation by means of bilinear method. These solutions can be transferred to those of Q3$_\delta$ equation in the Adler-Bobenko-Suris (ABS) list. Then making use of degeneration relation we obtain rational solutions for Q2, Q1$_\delta$, H3$_\delta$, H2 and H1. These rational solutions are in Casoratian form and the basic column vector satisfies an extended condition equation set.