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Structure of partial difference equations with continuous symmetries and conservation laws

Lead Research Organisation: University of Kent
Department Name: Sch of Maths Statistics & Actuarial Sci

Abstract

Abstracts are not currently available in GtR for all funded research. This is normally because the abstract was not required at the time of proposal submission, but may be because it included sensitive information such as personal details.
 
Description This join EPSRC project EP/I038675/1 (funding a research associate for three years hosted in the Applied Mathematics Department, University of Leeds) and EP/I038659/1 (funding a PhD student hosted in the University of Kent).

The achievements from this award can be demonstrated in the following aspects:

1. Working on the project the research team has produced 17 papers in which 13 are already published in high rank journals and 4 are submitted (all papers are available on arxive). Recently a PhD student (in Kent) has successfully defended his thesis which is based on the part of the project findings.

2. Together with S-Sh.Gao and C.M.Yuan we have made a considerable progress in the reformulation of the theory of integrable differential-difference and finite difference equations in terms of differential and difference algebra. This is a solid basis for future rigorous theory of integrable difference equations.

3. We found explicit necessary conditions for integrability of finite difference equations. First order conditions are suitable for study quadrilateral equations which include all equations from the Adler-Bobenko-Suris list. We explored the second order integrability conditions for difference equations and derived a new integrable system together with its symmetries, conservation laws and related differential-difference systems.

4. Working on the extension of the Lenard scheme we have developed a new approach called the ``O'' scheme, based on the indecomposable sl(2,C)-modules in the Bernstein-Gelfand-Gelfand category O, to construct master symmetries and further to generate infinite hierarchies of symmetries, conservation laws for integrable equations. The discovered correspondence between elements of the module and symmetry structures is a new promising direction of research.

5. We studied the link between curvature flows for polygon evolutions and integrable discrete equations and maps
using newly developed discrete moving frame method.

6. We have undertaken a systematic study of differential-difference and finite difference equations associated with finite reduction groups. We systemically investigated the elementary Darboux transformations of Lax representations
with finite reduction groups to study the connections among partial differential, differential-difference and finite difference systems as well as with integrable Yang-Baxter maps. Our construction can be naturally generalised to a non-commutative setting, such as Grassmann algebra valued differential difference and finite difference equations and Yang-Baxter maps.

7. Together with V. Sokolov we lifted the Drinfeld-Sokolov relation of integrable systems with Kac-Moody algebras to the level of differential-difference equations and Yang-Baxter maps.
Exploitation Route Differential-difference and finite-difference system emerging form the Darboux transformations of Lax operators can be regarded as integrable discretisations of the corresponding partial differential equations. These discretisations can be used for fast processing of optical signals in fibre optic communication (UK EPSRC Programme Grant UNLOC EP/J017582/1, ERC project ULTRALASER) and other numerical applications.

The basic framework and methods developed in this project have formed and will be continued to be a core material for postgraduate-level lecture courses at the EPSRC funded Taught Course Centres (LTCC and MAGIC).

Each of five publications have been cited over 10 times according to Google scholar. The research findings will be used and taken forward by researchers in both applied and pure mathematics.
Sectors Digital/Communication/Information Technologies (including Software)

Education
URL http://arxiv.org
 
Description This join EPSRC project EP/I038675/1 (funding a research associate for three years hosted in the Applied Mathematics Department, University of Leeds) and EP/I038659/1 (funding a PhD student hosted in the University of Kent). The PIs organised two-day workshop ``Algebraic Methods in Theory of Differential and Difference Equations'' on 6-7 December 2013 in the School of Mathematics, Statistics and Actuarial Science at the University of Kent (partially funded by the LMS). The list of attendees included established mathematicians, early career researchers and PhD students from various UK institutions (Essex, Glasgow, Kent, Leeds, Loughborough, Northumbria, Plymouth, Surrey, UCL) and from abroad (Bulgaria, France, Germany, Italy, Netherlands, Russia, South Africa, USA), with the total number of attendees being 43. The workshop brought together mathematical researchers working in two different, but closely related, fields: differential and difference algebra on the one hand and differential and difference equations on the other, and to further promote dialogue between them. In particular, the workshop programme was comprised of 17 talks, of which six were given by overseas speakers and four 20-minute talks were presented by female PhD students. The abstracts of all 17 talks are available to download from the website (http://www.kent.ac.uk /smsas/events/algebraic-methods.html). This workshop with a specific focus attracted more participants than we expected. It offered the attendees some viewpoints that were different from but clearly related to their own research. A short two days workshop ``Algebraic Interfaces of Integrability'' was held on 15-16 of May, 2015 at Leeds (partially supported by the LMS). Both PIs presented the results obtained from this research project. A complete list of talks with abstracts can be found at http://www1.maths.leeds.ac.uk/cnls/research/integrable/cqi/2015/sol15.html The basic framework and methods developed in this project formed a core material for postgraduate-level lecture courses at the EPSRC funded Taught Course Centres (London and MAGIC TCC). The 10-hour advanced LTCC course on ``Symmetries, Conservation Laws and the Variational Complex'' was given by J.P. Wang and 20-hour MAGIC course on ``Integrable Systems'' was given by A.V. Mikhailov in the spring term of 2014 and 2016. The research team has published 13 original papers in leading international scientific journals, such as Letters in Mathematical Physics, nonlinearity, Physica D, Journal of Mathematical Physics, Studies in Applied Mathematics, Theoretical and Mathematical Physics and Journal of Physics A, and has submitted 4 more articles. All papers are distributed using electronic preprint archives available on the Internet: http://arxiv.org/. The research finding were presented not only at international colloquia, workshops and conferences but also delivered on the International Congress on Industrial and Applied Mathematics (outside of the integrable systems community). Recent promising developments in the optic communications based on spectral transform and fast algorithms for solution of differential-difference systems, which already enables to increase the capacity of the fibre optic communication systems by 15% and beyond (UK EPSRC Programme Grant UNLOC EP/J017582/1, ERC project ULTRALASER). The differential-difference systems and finite difference systems which we study in our project have immediate applications to engineering and technological problems which have considerable impact to the price and quality of telecommunication.
Sector Digital/Communication/Information Technologies (including Software),Education
Impact Types Cultural

Economic
 
Description LTCC
Geographic Reach National 
Policy Influence Type Influenced training of practitioners or researchers
Impact 10 hour LTCC (the London Taught Course Centre) advanced course for PhD students on Symmetries, Conservation Laws and the Variational Complex were developed and delivered in 2015. Six PhD students completed the final exercise after following all lectures.
URL http://www.ltcc.ac.uk/courses/all_course_list.php
 
Description Scheme 1 Conference grant
Amount £4,130 (GBP)
Funding ID 11310 
Organisation London Mathematical Society 
Sector Academic/University
Country United Kingdom
Start 12/2013 
End 12/2013
 
Description Scheme 4 Research in Pairs
Amount £600 (GBP)
Funding ID 41418 
Organisation London Mathematical Society 
Sector Academic/University
Country United Kingdom
Start 03/2015 
End 04/2015
 
Description Masterclass for GCSE students 
Form Of Engagement Activity A formal working group, expert panel or dialogue
Part Of Official Scheme? Yes
Geographic Reach Local
Primary Audience Schools
Results and Impact The workshop sparked the interests in Mathematics.

After the workshop, the GCSE students took up the summer projects funded by the Nuffield fundation.
Year(s) Of Engagement Activity 2013
URL http://www.kent.ac.uk/smsas/outreach/masterclasses.html