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Perturbative symmetry approach for differential-difference equations

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posted on 2022-07-01, 14:00 authored by Alexander Mikhailov, Vladimir NovikovVladimir Novikov, Jing Ping Wang

We propose a new method to tackle the integrability problem for evolutionary differential–difference equations of arbitrary order. It enables us to produce necessary integrability conditions, to determine whether a given equation is integrable or not, and to advance in classification of integrable equations. We define and develop symbolic representation for the difference polynomial ring, difference operators and formal series. In order to formulate necessary integrability conditions, we introduce a novel quasi-local extension of the difference ring. We apply the developed formalism to solve the classification problem of integrable equations for anti-symmetric quasi-linear equations of order (−3,3) and produce a list of 17 equations satisfying the necessary integrability conditions. For every equation from the list we present an infinite family of integrable higher order relatives. Some of the equations obtained are new.

Funding

A novel approach to integrability of semi-discrete systems

Engineering and Physical Sciences Research Council

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Exact solutions for discrete and continuous nonlinear systems

Engineering and Physical Sciences Research Council

Find out more...

Exact solutions for discrete and continuous nonlinear systems

Engineering and Physical Sciences Research Council

Find out more...

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Communications in Mathematical Physics

Volume

393

Issue

2

Pages

1063 - 1104

Publisher

Springer (part of Springer Nature)

Version

  • VoR (Version of Record)

Rights holder

© The Authors

Publisher statement

This is an Open Access Article. It is published by Springer under the Creative Commons Attribution 4.0 International Licence (CC BY). Full details of this licence are available at: https://creativecommons.org/licenses/by/4.0/

Acceptance date

2022-03-17

Publication date

2022-05-03

Copyright date

2022

ISSN

0010-3616

eISSN

1432-0916

Language

  • en

Depositor

Dr Vladimir Novikov. Deposit date: 21 March 2022

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