The course is an introduction to the theory of integrable systems. We will start with finite-dimensional Hamiltonian systems, Liouville integrability, and the concept of a Lax pair. These will be illustrated by the celebrated examples of Calogero-Moser and Toda systems. After that we will move on to infinite-dimensional systems, starting with KdV and KP equations. We will discuss the corresponding hierarchies in Lax form, and special classes of solutions (solitons). This will allow us to see a connection with the Calogero-Moser particles. The second part of the course will look more deeply into integrability of various types of nonlinear partial differential equations, differential-difference and partial difference equations. By integrability in these cases we understand 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.
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.
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.
 Ablowitz, M.J. Clarkson P.A. 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering, CUP.
 Ablowitz, M.J. and Segur, H. 1981 Solitons and the Inverse Scattering Transform, SIAM.
 Dodd, R.K., Eilbeck, J.C., Gibbon, J.D. and Morris, H.C. 1982 Solitons and Nonlinear Waves Equations, Academic Press, Inc.
 Mikhailov, A.V. (Ed) 2009 Integrability, Springer.
 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.
 Newell, A.C. 1985 Solitons in Mathematics and Physics, SIAM.
 Zakharov, V.E.(Ed) 1991 What is Integrability? Springer.