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2018/19 Undergraduate Module Catalogue

MATH3491 Discrete Systems and Integrability

15 creditsClass Size: 50

Module manager: Professor Frank Nijhoff

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2018/19

Pre-requisite qualifications

MATH2375 or equivalent.

This module is mutually exclusive with

MATH5492MAdvanced Discrete Systems and Integrability

This module is approved as a discovery module

Module summary

This module gives an overview of the modern theory of the integrability of discrete systems and of difference equations, and highlights its many intriguing connections with other areas in mathematics, such as the theory of special functions, algebra and (discrete) geometry, and with physics.


On completion of this module, students should be able to:
a) construct simple solutions of ordinary and partial difference equations (P-Es);
b) use B├Ącklund 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.


In the last decade 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
- B├Ącklund transformations
- Lax pairs and conservation laws
- Discrete solitons
- Integrable dynamical mappings and their invariants
- Connections to the theory of special functions.

Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Delivery typeNumberLength hoursStudent hours
Private study hours117.00
Total Contact hours33.00
Total hours (100hr per 10 credits)150.00

Private study

Studying and revising of course material.
Completing of assignments and assessments.

Opportunities for Formative Feedback

Regular example sheets.

Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Exam typeExam duration% of formal assessment
Standard exam (closed essays, MCQs etc)2 hr 30 mins100.00
Total percentage (Assessment Exams)100.00

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: 20/03/2018


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