Symmetries and Integrability of Differential Equations

Annotation

The course  introduces students to the concepts of symmetries and integrability of differential equations using as examples a number of integrable systems which often show up in various areas of mathematics and theoretical physics.  A particular attention is paid to hidden rich algebraic and/or analytic structures associated with integrable equations.  Topics covered include 

Lax representation, Bi-Hamiltonian formalism, and symmetry approach to classification of integrable evolution equations.

Course plan

1. Introduction ``What is integrability?'' Examples of integrable systems.
2. Differential equation as a submanifold of the jet space. Vector fields. Total derivatives and symmetries.
3. Point and contact transformations. Differential substitutions of Miura type. Integrability in quadratures. Lie theorem.
4. Polynomial Poisson brackets. Bi-Hamiltonian formalism and shift argument method.  
5. Lax representation for ODEs. Euler top. Integrable cases in rigid body dynamics. 
6. Analytical properties of general solutions of ODEs. Painleve approach to integrability. Painleve equations.
7. Korteweg-de Vries  equation: conservation laws, Lax pair, Darboux transformations, solitonic solutions. 
8. Algebra of pseudo-differential series. Hierarchy of KdV equation: higher symmetries and recursion operator. 
9. Matrix Lax pairs. Non-linear Schroedinger equation: symmetries and conservation laws.
10. Lie algebras and integrable systems.
11. Symmetry approach to classification of integrable evolution equations.
12. Polynomial multi-component integrable systems and non-associative algebras.
13. Darboux integrable hyperbolic equations. Liouville equation and Toda lattices.

Literature

1. Olver P.J., Applications of Lie Groups to Differential Equations, (2nd edn), Graduate
2. Texts in Mathematics, 1993, 107, Springer–Verlag, New York.
3. Ibragimov N. H.,  Transformation Groups Applied to Mathematical Physics, Dordrecht: D. Reidel, 1985, 394 pp. 
4. Newell A.,  Solutions in Mathematics and Physics,  SIAM, Philadelphia 1985, 244 pp. 
5. Sokolov V. V., Algebraic structures in Integrability, World Scientific, Singapore, 2020,
6. ISBN 978-981-121-966-5, 321 pp.