Geometrical Theory of Nonlinear Differential Equations

Annotation

The course introduces basic concepts and methods of the formal theory of differential equations: geometry of jet spaces, algebras of differential operators, Lie pseudogroups, characteristics, prolongations and symbols, compatibility theory, symmetries and conservation laws, exact solutions. The main theorems are proved and key examples are considered.

The emphasis is on the algebra-geometric properties of equations of mathematical physics. Affine structure on fibers of jet bundles leads to basic tools such as evolutionary differentiation and the linearization operator. The concept of symmetry will be central to the course, related to conservation laws through the Noether theorem. We finish with variational bicomplex and applications to integrable systems.

Course plan

1. The space of jets of mappings, sections and submanifolds. Cartan distribution on finite and infinite jets. Contact geometry.
2. Automorphisms of jet spaces: Lie-Backlund theorem. Integral submanifolds. Intrinsic and extrinsic automorphisms. Prolongations.
3. Linear systems and D-modules. Nonlinear systems, linearization on a background solution. Operators in total derivatives.
4. Compatibility of overdetermined systems. Involution. The Cartan-Kahler theorem. The space of solution and its functional dimension.
5. Applications to geometric structures, curvature. The method of Cartan. G-structures and the Sternberg continuation. Examples: Riemannian, conformal, projective structures.
6. Classical symmetries of differential equations. Calculations using symbolic packages Maple/Mathematica. Examples.
7. Higher symmetries of differential equations. Examples: Korteweg–de Vries and Kadomtsev-Petviashvili equations. Dispersionless limit.
8. Lie algebras of vector fields and Lie pseudogroups. Differential invariants and differential constraints. Application: exact solutions. 
9. Symmetries of geometric structures. Calculations for flat and curved structures.
10. Introduction to the calculus of variations and variational bicomplex. Invariant Lagrangians and invariant Euler-Lagrange equations. Gauge-invariant equations.

Literature

1. I. S. Krasil'shchik, A. M. Vinogradov (Eds). Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Translations of Mathematical Monographs, AMS, 1999.
2. P. Olver. Applications of Lie Groups to Differential Equations. Graduate texts in mathematics, Springer, 1986. P. Olver. Equivalence, Invariants and Symmetry. Cambridge University Press. 1995.
3. I. Anderson, Variational Bicomplex (unpublished book for InterScience Press). https://ncatlab.org/nlab/files/AndersonVariationalBicomplex.pdf.
4. B. Kruglikov, V. Lychagin. Geometry of Differential equations; In: Handbook on Global Analysis D.Krupka and D.Saunders Eds., 725-771, 1214, Elsevier Sci. 2008. http://preprints.ihes.fr/2007/M/M-07-04.pdf.