# The 1st summer school on geometry of differential equations

# Lecture Notes for the Advanced Course: pdf

# Notes for an exercise from the Basic Course: pdf

# Original Announcement

The summer school will take place **17-21 September 2012** and will be held in the Institute of Mathematics of Silesian University in Opava, Czech Republic.

The school will be hosted in Davidův mlýn, see the location on Google Maps

**Applications** should be sent to: **school-gde@math.slu.cz**

There is no formal deadline for applications, but please contact the organizers **as soon as possible**, because of limited capacity of the school.

There will be two parallel courses.

## Basic Course: Symmetry and conservation laws

(Raffaele Vitolo, University of Salento, and G. Moreno, Silesian University in Opava)

The purpose of this course is to introduce the students to the geometrical theory of differential equations. Usually, differential equations are regarted as equations involving an unknown function together with its (partial) derivatives up to some order. According with the geometric viewpoint, equations shall be considered as submanifolds of a space whose coordinates represent derivatives. Such a space is endowed with the *Cartan distribution*, i.e., a linear subspace of tangent vectors at each point. Solutions of a given equation are particular submanifolds of the equation which are tangent to the Cartan distribution.

The above viewpoint allows us to use the wide spectrum of powerful tools and ideas from differential geometry in order to study differential equations. This led to a deeper understanding of many old methods of solution to differential equations and many new applications.

In this 40 hours course, after a brief recall of differential-geoemtric prerequisites, we shall describe a minimal but satisfactory picture of the modern theory of symmetries and conservation laws of (nonlinear) differential equations, and show computational examples, also using computer algebra systems.

The morning lessons will have theoretical character, while in the afternoon students will solve problems which will enable them step-by-step to elementary computations of symmetries and conservation laws. It is our main concern that all student gain some important theoretical ideas as well as concrete skills.

*Syllabus*:

**Monday, September 17**

Morning session

*Preliminaries* (G.M., 2h) Manifolds, vector fields, differential forms, Lie derivatives, elementary theory of distributions and Frobenius' Theorem.

*Jet Spaces* (R.V., 2h) Definition of jet space and its Cartan distribution; elementary properties.

Afternoon session

*Exercises* (G.M.+R.V., 4h) Exercises on jet spaces and the Cartan distribution.

**Tuesday, September 18**

Morning session

*Symmetries of jet spaces* (R.V., 2h) Finite and infinitesimal symmetries of jet spaces and their Cartan distribution.

*Differential equations* (G.M., 2h) Ordinary differential equations, ([1], Chap. 1), first order differential equations, ([1], Chap. 2), general differential equations ([1], Chap. 3) and their Cartan distribution.

Afternoon session

*Exercises* (G.M.+R.V., 4h) Advanced exercises on jet spaces; initial exercises on differential equations.

**Wednesday , September 19**

Morning session

*Symmetries of differential equations* (G.M., 2h) Symmetries as vector fields tangent to a PDE, generating functions ([1], Sec. 3.6).

*Infinite Jet Spaces* (R.V., 2h) Definition, vector fields on infinite order jets, Cartan distribution and its infinitesimal automorphism, evolutionary derivations, Jacobi bracket, linearization.

Afternoon session

*Computer experiments*: Students will be brought to a laboratory of the Silesian University in Opava to learn the program **Jets**, a set of Maple procedures for computations in jet spaces. Exercises on the theoretical part will be shown in details.

**Thursday , September 20**

Morning session

*Infinitely Prolonged PDEs and Their Higher Symmetries* (G.M., 2h) ([1], Sec. 4.3)

*Applications of symmetries* (R.V., 2h) Invariant solutions, reduction of a differential equation via symmetries, examples of computations.

Afternoon session

*Exercises* (G.M.+R.V., 4h) Advanced exercises on computation of symmetries and higher symmetries.

**Friday , September 21**

Morning session

*Characteristics of PDEs* (G.M., 2h) The vertical geometry of PDEs, singular integral elements of codimension one, the equation of singularities of solutions of a PDE, characteristics of a PDE. Computation of contact invariants by means of a canonical distribution on the Grassmannian of involutive integral elements (see [2]).

*Conservation Laws* (R.V., 2h) Conservation laws, trivial
conservation laws, symmetries and conservation laws.

Afternoon session

*Computer experiments*: Advanced computations with *Jets*: higher symmetries and conservation laws.

**Bibliography**

[1] A. V. Bocharov, V. N. Chetverikov, S.V. Duzhin, N.G. Khorkova, I.S. Krasilshchik, A.V. Samokhin, Yu.N. Torkhov, A.M. Verbovetsky and A. M. Vinogradov: Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, I.S. Krasilshchik and A. M. Vinogradov eds., Translations of Math. Monographs 182, Amer. Math. Soc. (1999), Russain original.

[2] R.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmidt, and P.A. Griffiths. Exterior differential systems, volume 18 of Mathematical Sciences Research Institute Publications. Springer-Verlag, New York, 1991, pdf

## Advanced Course: Recursion operators

(Joseph Krasil'shchik, Silesian University in Opava and Moscow Independent University, and Alexander Verbovetsky, Moscow Independent University)

*Syllabus*:

- Introduction: finite and infinite jets, the Cartan distribution, equations and prolongations. Solutions.
- Symmetries. Their description via generating functions. The Jacobi bracket. Linearization (the Frechet derivative) and defining equation for symmetries.
- Examples: the Burgers and Korteweg-de Vries (KdV) equations.
- An application: 1-solitons as invariant solutions of the KdV equation.
- Back to Item 3 how to finish the computation of symmetries for the KdV equation. Lenard's recursion operator.
- What happens when applying Lenard's recursion operator to -dependent symmetries? A
*naïve*way to introduce nonlocal variables. Other examples of nonlocal constructions. - Geometrization: coverings. More examples. Relation to conservation laws. Zero-curvature representations.
- The Wahlquist-Estabrook algebra.
- Examples of computations: the Burgers and KdV equations. The Cole-Hopf and Miura transformations.
- An application: Bäcklund transformations. Multi-soliton solutions of the KdV equation. Multi-kink solutions of the sine-Gordon equation. Bianchi's permutability theorem.
- Back to Item 6 what did we obtain using Lenard's operator? Nonlocal symmetries and shadows.
- The reconstruction theorem. The universal Abelian covering. Master-symmetries and the commutator problem.
- Back to recursion operators: The tangent covering . Symmetries as its holonomic sections. Geometric viewpoint on recursion operators: auto-Bäcklund transformations of . Examples.
- An alternative viewpoint: shadows of symmetries in . Examples. "Canonical" conservation laws in the tangent covering.
- Commutativity of hierarchies: hereditary recursion operators and the Nijenhuis torsion.
- Perspectives.
- Recursion operators for a class of multidimensional dispersionless PDEs: an explicit construction

The basic course is aimed at the beginners, with the pace and style of presentation to match. The advanced course is aimed at the students who are already familiar with the contents of the basic course.

The courses will provide students with a comprehensive presentation of the respective subjects, and introduce them to the basic motivations, methods and results of the relevant field of study. The participants will also be informed about the open problems in the field.

The courses will include training sessions, in the course of which the participants will learn to use the software for the computer-aided calculation of symmetries and conservation laws, resp. of coverings and recursion operators.

The Summer School will last for five days with a total of 40 academic hours of lectures and training sessions.

The teaching will be in English and will take place in the form of lectures in the morning and training sessions in the afternoon.

In the course of the training sessions the participants will solve (sets of) problems given to them and submit their solutions to the instructor who will provide advice and feedback on these if need be.

The successful participants will receive a certificate; the latter will be awarded on the basis of performance at the training sessions.

**Participation in the school is free of charge.**

**Travel and subsistence costs are to be covered by the participants themselves.**

This School is the first of three summer schools on the geometry of differential equations supported by the European Social Fund under the project CZ.1.07/2.3.00/20.0002.

Two further summer schools on similar subjects will take place in 2013 and 2014.