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.
At level 7, topics will be studied and assessed to greater depth.
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 7 module students will be able to:
1 demonstrate systematic understanding of integrable systems;
2 demonstrate the capability to solve complex problems using a very good level of skill in calculation and manipulation of the material in the following areas: nonlinear
differential equations, Hamiltonian systems, nonlinear difference equations;
3 apply a range of concepts and principles in integrable systems in various different contexts, showing good judgment in the selection and application of tools and
techniques.
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