Algebraic Methods in Physics: M7022M

in combination with

Methods for Integrating Nonlinear Differential Equations: MAM013F

Period 3: 2011

Lecturer an Examiner: Prof. Norbert Euler, Department of Mathematics, LTU
Email: norbert@ltu.se

Duration of Course:
LP3: January - March, 2011.

Contents:
The course aims to provide a general introdution to symmetry analysis of differential equations and will concentrate on the practical matters of how to derive and use symmetries of differential equations in order to calculate exact solutions and first integrals or conservation laws of nonlinear ordinary differential equations (ODEs) and nonlinear partial differential equations (PDEs). We'll also discuss some of the more recent developments regading nonlocal transformations between differential equations (for ODEs and PDEs) and the related classifcation problems, as well as some aspects of recursion operators for Lie-Baecklund symmetries of ODEs and PDEs and integrable hierarchies of differential equations.

Since this course consists of only ten lectures and given the wide range of subjects which we aim to include, we will not be able to discuss the deeper mathematical significance of some of the statements for the symmetry calculations. References will however be provided for the interested reader.

The course can be devided into the following main sections:

Lie transformation groups and Lie group generators.
Prolongations and Lie algebras.
Lie point symmetries: Invariance of the differential equations.
Applications to ODEs.
Applications to PDEs.
Lie-Baecklund symmetries, recursion operators and integrability.
Transformations between differential equations.

Literature:
Complete Lecture Notes will be provided free of charge.

Additional books on this subject:

Symmetry Methods for Differential Equations: A Beginner's Guide
P. E. Hydon (Cambridge University Press - 2000)
Symmetries of Differential Equations
G. W. Bluman and S. Kumei (Springer-Verlag - 1989)
Symmetries and Integration Methods for Differential Equations
G. W. Bluman and S. C. Anco (Springer-Verlag - 2002)
Applications of Lie Groups to Differential Equations
P. J. Olver (Springer-Verlag - 1st edition 1986 and 2nd edition 1993)

Credit System:
Exercises will be provided during class. Your solutions will determine your grades for this course.

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