Nonlinear Symmetries and Nonlinear Equations

✍ Scribed by Giuseppe Gaeta (auth.)

• Publisher: Springer Netherlands
• Year: 1994
• Tongue: English
• Leaves: 274
• Series: Mathematics and Its Applications 299
• Edition: 1
• Category: Library

✦ Synopsis

The study of (nonlinear) dift"erential equations was S. Lie's motivation when he created what is now known as Lie groups and Lie algebras; nevertheless, although Lie group and algebra theory flourished and was applied to a number of dift"erent physical situations -up to the point that a lot, if not most, of current fun­ damental elementary particles physics is actually (physical interpretation of) group theory -the application of symmetry methods to dift"erential equations remained a sleeping beauty for many, many years. The main reason for this lies probably in a fact that is quite clear to any beginner in the field. Namely, the formidable comple:rity ofthe (algebraic, not numerical!) computations involved in Lie method. I think this does not account completely for this oblivion: in other fields of Physics very hard analytical computations have been worked through; anyway, one easily understands that systems of dOlens of coupled PDEs do not seem very attractive, nor a very practical computational tool.

✦ Table of Contents

Front Matter....Pages i-xix
Geometric setting....Pages 1-22
Symmetries and their use....Pages 23-44
Examples....Pages 45-54
Evolution equations....Pages 55-82
Variational problems....Pages 83-95
Bifurcation problems....Pages 97-121
Gauge theories....Pages 123-154
Reduction and equivariant branching lemma....Pages 155-173
Further Developements....Pages 175-204
Equations of Physics....Pages 205-222
Back Matter....Pages 223-260

✦ Subjects

Ordinary Differential Equations;Partial Differential Equations;Theoretical, Mathematical and Computational Physics