MATH5492M Advanced Discrete Systems and Integrability
20 Credits Class Size: 25
Module manager: Professor Frank Nijhoff
Email: F.W.Nijhoff@leeds.ac.uk
Taught: Semester 1 (Sep to Jan) View Timetable
Year running
2018/19
Pre-requisite qualifications
MATH2375 or equivalent.
Mutually Exclusive
| 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 |
44 |
| Private study hours |
156 |
| Total Contact hours |
44 |
| Total hours (100hr per 10 credits) |
200 |
Opportunities for Formative Feedback
Regular example sheets.
Exams
| Exam type |
Exam duration |
% of formal assessment |
| Standard exam (closed essays, MCQs etc) |
3.0 Hrs Mins |
100 |
| Total percentage (Assessment Exams) |
100 |
Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated
Reading List
The reading list is available from the Library website
Last updated: 3/20/2018
Errors, omissions, failed links etc should be notified to the Catalogue Team