On some new forms of lattice integrable equations

TL;DR

This paper introduces new integrable differential-discrete equations related to KdV, mKdV, and Sine-Gordon, providing novel discretizations and connections to known equations like lattice mKdV and Tzitzeica.

Contribution

It presents new integrable discretizations of delay-Painleve inspired systems, including higher order versions, and establishes their relation to classical equations via Miura transformations.

Findings

New integrable discretizations of KdV, mKdV, and Sine-Gordon systems.

Identification of the lattice mKdV as a discretization of the intermediate sine-Gordon.

All new lattice equations can be reduced to QRT mappings.

Abstract

Inspired by the forms of delay-Painleve equations, we consider some new differential-discrete systems of KdV, mKdV and Sine-Gordon - type related by simple one way Miura transformations to classical ones. Using Hirota bilinear formalism we construct their new integrable discretizations, some of them having higher order. In particular, by this procedure, we show that the integrable discretization of intermediate sine-Gordon equation is exactly lattice mKdV and also we find a bilinear form of the recently proposed lattice Tzitzeica equation. Also the travelling wave reduction of these new lattice equations is studied and it is shown that all of them, including the higher order ones, can be integrated to Quispel-Roberts-Thomson (QRT) mappings.