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2018/19 Taught Postgraduate Module Catalogue

MATH5492M Advanced Discrete Systems and Integrability

20 creditsClass Size: 25

Module manager: Professor Frank Nijhoff
Email: F.W.Nijhoff@leeds.ac.uk

Taught: Semester 1 View Timetable

Year running 2018/19

Pre-requisite qualifications

MATH2375 or equivalent.

This module is mutually exclusive with

 MATH3491 Discrete Systems and Integrability

This module is approved as an Elective

Module summary

This module aims at giving an overview of the modern theory and at highlighting its many intriguing connections with other areas in mathematics, such as the theory of special functions, algebra and (discrete) geometry, and with physics.

Objectives

On completion of this module, students should be able to:
a) construct simple solutions of ordinary and partial difference equations;
b) use Bäcklund transformations to obtain discrete equations from continuous ones and vice versa;
c) manipulate Lax pairs and overdetermined systems of linear difference equations;
d) derive continuum limits from integrable difference equations;
e) perform computations associated with soliton solutions;
f) derive integrable mappings from lattice equations and the corresponding invariants;
g) use addition formulae for elliptic functions to parametrise solutions of difference equations;
h) construct similarity reductions for integrable partial differential equations and derive discrete Painlevé equations.

Syllabus

In the last two decades the integrability of discrete systems and of difference equations has gained a lot of attention. These systems can manifest themselves in various ways: as discrete dynamical systems (mappings), as nonlinear ordinary difference equations (including analytic difference equations), as recurrence relations for orthogonal polynomials, and as lattice equations (ie partial difference equations).

What is striking is that these systems exhibit quite similar properties as their continuous analogues which are integrable ODEs, evolutionary dynamical systems or nonlinear evolution equations of soliton type. However, it seems that the theory of discrete systems is even richer and many of its key features have only been discovered rather recently.

Topics include:
- Lattice equations and their continuum limits
- Bäcklund transformations
- Lax pairs and conservation laws
- Discrete solitons
- Similarity reduction
- Integrable dynamical mappings and invariants
- Discrete Painlevé equations
- Addition formulae for elliptic functions
- Difference equations and connections to the theory of special functions.

Teaching methods

 Delivery type Number Length hours Student hours Lecture 44 1.00 44.00 Private study hours 156.00 Total Contact hours 44.00 Total hours (100hr per 10 credits) 200.00

Opportunities for Formative Feedback

Regular example sheets.

Methods of assessment

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 3 hr 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated