This module is not currently running in 2024 to 2025.
Integrable systems are special dynamical systems which can be solved exactly in some sense. They arise in a variety of settings, ranging from Hamiltonian systems and nonlinear wave equations to difference equations. This module covers the origins of the subject as well as modern topics like integrable maps and lattice equations.
* Liouville integrability in classical mechanics. Hamiltonian mechanics. Canonical symplectic form and Poisson brackets. Liouville's theorem (statement and examples). Lax pairs for finite-dimensional systems.
* Soliton equations. History and physical origins (e.g. Korteweg-de Vries and/or sine-Gordon). Conservation laws. Hamiltonian formalism. Lax pairs.
* Construction of solitons. Introduction to inverse scattering. Darboux-Bäcklund transformations. Hirota's method.
* Discrete integrability. Symplectic maps. Liouville's theorem (discrete version). Integrable lattice equations. Discrete Lax pairs with examples.
Total contact hours: 42
Private study hours: 108
Total study hours: 150
Level 6:
Assessment 1 (10-15 hrs) 20%
Assessment 2 (10-15 hrs) 20%
Examination (2 hours) 60%
Level 7:
Assessment 1 (10-15 hrs) 20%
Assessment 2 (10-15 hrs) 20%
Examination (2 hours) 60%
Reassessment methods
Like-for-like
O. Babelon, D. Bernard and M. Talon, Introduction to Classical Integrable Systems, Cambridge Monographs on Mathematical Physics, Cambridge University Press, 2003.
M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Note Series 149, Cambridge University Press, 1992.
P.G. Drazin and R.S. Johnson, Solitons: an introduction, Cambridge Texts in Applied Mathematics 2, Cambridge University Press, 1989.
J. Hietarinta, N. Joshi and F. W. Nijhoff, Discrete Systems and Integrability, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2016.
See the library reading list for this module (Canterbury)
The intended subject specific learning outcomes:
On successfully completing the level 6 module students will be able to:
1 demonstrate systematic understanding of key aspects of integrable systems;
2 demonstrate the capability to deploy established approaches accurately to analyse and solve problems using a reasonable level of skill in calculation and manipulation of
the material in the following areas: nonlinear differential equations, Hamiltonian systems, nonlinear difference equations;
3 apply key aspects of integrable systems in well-defined contexts, showing judgement in the selection and application of tools and techniques.
The intended generic learning outcomes:
On successfully completing the level 6 module students will be able to:
1 manage their own learning and make use of appropriate resources;
2 understand logical arguments, identifying the assumptions made and the conclusions drawn;
3 communicate straightforward arguments and conclusions reasonably accurately and clearly;
4 manage their time and use their organisational skills to plan and implement efficient and effective modes of working;
5 solve problems relating to qualitative and quantitative information;
6 make competent use of information technology skills such as online resources (Moodle) and internet communication;
7 communicate technical material competently;
8 demonstrate an increased level of skill in numeracy and computation;
9 demonstrate the acquisition of the study skills needed for continuing professional development.
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