# MAGIC083: Integrable Systems

## Course details

A core MAGIC course

### Semester

Spring 2020
Monday, January 20th to Friday, March 27th

### Hours

Live lecture hours
20
Recorded lecture hours
0
80

### Timetable

Wednesdays
13:05 - 13:55 (UK)
Fridays
12:05 - 12:55 (UK)

## Description

The course is an introduction to the theory of integrable systems. We will consider mainly the infinite-dimensional systems such as nonlinear partial differential equations, differential-difference and partial difference equations. By integrability we understand the existence of an infinite hierarchy of symmetries and/or conservation laws. Lax representations are sufficient conditions for integrability. Corresponding Darboux transformations provide a link between integrable partial differential, differential-difference and partial difference equations. They enable us to construct exact multisoliton solutions, hierarchies of symmetries and conservation laws, as well as recursion operators. We will derive necessary conditions for integrability and apply them to the problem of classification of integrable systems. Main examples include nonlinear Schrödinger type equations, Volterra and Toda lattices, partial difference Boussinesq and Tzitzeica type equations. The major part of the course is based on well established theory, although some open yet unsolved problems and possible directions of research will also be presented.

### Prerequisites

Linear algebra and some elementary calculus. Excellent companions to the course Integrable Systems MAGIC083 are the MAGIC courses: Integrable Systems MAGIC067, Lie Groups and Lie Algebras MAGIC008, Nonlinear Waves MAGIC021.

### Syllabus

Systems of ordinary differential equations, vector fields, first integrals, symmetries. Theorem of S.Lie on integration in quadratures.
Partial differential equations, vector fields, symmetries, local conservation laws. Recursion operator. Symmetry reductions. Examples: KdV, NLS.
Lax representations for PDEs. Derivation of hierarchies of conservation laws and symmetries. Construction of the recursion operator. Construction of exact "soliton" solutions, Darboux and Bäcklund transformations. Example: NLS.
A chain of Bäcklund transformations as an integrable differential-difference system. Symmetries and local conservation laws of differential-difference systems. Example:Toda lattice.
Bianchi commutativity of Darboux transformations and integrable systems of partial difference equations. Symmetries and local conservation laws of partial difference equations. Example: NLS.
Formal pseudo-differential series residues and Adler's Theorem. Symmetries and/or conservation laws imply the existence of a formal recursion operator. Canonical conservation laws as integrability conditions for PDEs. Example: simple classification problem.
Generalisation to differential-Difference and partial difference cases. Integrability conditions. Examples:Volterra lattice, partial difference Boussinesq and Tzitzeica type equations.
BOOKS: [1] Ablowitz, M.J. Clarkson P.A. 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering, CUP. [2] Ablowitz, M.J. and Segur, H. 1981 Solitons and the Inverse Scattering Transform, SIAM. [3] Dodd, R.K., Eilbeck, J.C., Gibbon, J.D. and Morris, H.C. 1982 Solitons and Nonlinear Waves Equations, Academic Press, Inc. [4] Mikhailov, A.V. (Ed) 2009 Integrability, Springer. [5] Novikov, S.P., Manakov, S.V., Pitaevskii, L.P. and Zakharov, V.E. 1984 The Theory of Solitons: The Inverse Scattering Method, Consultants, New York. [6] Newell, A.C. 1985 Solitons in Mathematics and Physics, SIAM. [7] Zakharov, V.E.(Ed) 1991 What is Integrability? Springer.

## Lecturer

• AM

### Professor Alexander Mikhailov

University
University of Leeds

## Bibliography

### Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

## Assessment

The assessment for this course will be released on Monday 20th April 2020 at 00:00 and is due in before Monday 4th May 2020 at 11:00.

The assessment of the course will be in the form of open book exam. The set of questions will be based on the material which have been actually presented on the lectures and recorded.

Please note that you are not registered for assessment on this course.