Front Matter
Part I Local Transformation Groups
1 Preliminaries
1.1 Changes of frames of reference and point transformations
1.1.1 Translations
1.1.2 Rotations
1.1.3 Galilean transformation
1.2 Introduction of transformation groups
1.2.1 Definitions and examples
1.2.2 Different types of groups
1.3 Some useful groups
1.3.1 Finite continuous groups on the straight line
1.3.2 Groups on the plane
1.3.3 Groups in IRn
Exercises to Chapter 1
2 One-parameter groups and their invariants
2.1 Local groups of transformations
2.1.1 Notation and definition
2.1.2 Groups written in a canonical parameter
2.1.3 Infinitesimal transformations and generators
2.1.4 Lie equations
2.1.5 Exponential map
2.1.6 Determination of a canonical parameter
2.2 Invariants .
2.2.1 Definition and infinitesimal test
2.2.2 Canonical variables
2.2.3 Construction of groups using canonical variables
2.2.4 Frequently used groups in the plane
2.3 Invariant equations
2.3.1 Definition and infinitesimal test
2.3.2 Invariant representation of invariant manifolds
2.3.3 Proof of Theorem 2.9
2.3.4 Examples on Theorem 2.9
Exercises to Chapter 2
3 Groups admitted by differential equations
3.1 Preliminaries
3.1.1 Differential variables and functions
3.1.2 Point transformations
3.1.3 Frame of differential equations
3.2 Prolongation of group transformations
3.2.1 One-dimensional case
3.2.2 Prolongation with several differential variables
3.2.3 General case
3.3 Prolongation of group generators
3.3.1 One-dimensional case
3.3.2 Several differential variables
3.3.3 General case
3.4 First definition of symmetry groups
3.4.1 Definition
3.4.2 Examples
3.5 Second definition of symmetry groups
3.5.1 Definition and determining equations
3.5.2 Determining equation for second-order ODEs
3.5.3 Examples on solution of determining equations
Exercises to Chapter 3
4 Lie algebras of operators
4.1 Basic definitions
4.1.1 Commutator
4.1.2 Properties of the commutator
4.1.3 Properties of determining equations
4.1.4 Lie algebras
4.2 Basic properties
4.2.1 Notation
4.2.2 Subalgebra and ideal
4.2.3 Derived algebras
4.2.4 Solvable Lie algebras
4.3 Isomorphism and similarity
4.3.1 Isomorphic Lie algebras
4.3.2 Similar Lie algebras
4.4 Low-dimensional Lie algebras
4.4.1 One-dimensional algebras
4.4.2 Two-dimensional algebras in the plane
4.4.3 Three-dimensional algebras in the plane
4.4.4 Three-dimensional algebras in IR3
4.5 Lie algebras and multi-parameter groups
4.5.1 Definition of multi-parameter groups
4.5.2 Construction of multi-parameter groups
Exercises to Chapter 4
5 Galois groups via symmetries
5.1 Preliminaries
5.2 Symmetries of algebraic equations
5.2.1 Determining equation
5.2.2 First example
5.2.3 Second example
5.2.4 Third example
5.3 Construction of Galois groups
5.3.1 First example
5.3.2 Second example
5.3.3 Third example
5.3.4 Concluding remarks
Assignment to Part I
Part II Approximate Transformation Groups
6 Preliminaries
6.1 Motivation
6.2 A sketch on Lie transformation groups
6.2.1 One-parameter transformation groups
6.2.2 Canonical parameter
6.2.3 Group generator and Lie equations
6.2.4 Exponential map
6.3 Approximate Cauchy problem
6.3.1 Notation
6.3.2 Definition of the approximate Cauchy problem
7 Approximate transformations
7.1 Approximate transformations defined
7.2 Approximate one-parameter groups
7.2.1 Introductory remark
7.2.2 Definition of one-parameter approximate transformation groups
7.2.3 Generator of approximate transformation group
7.3 Infinitesimal description
7.3.1 Approximate Lie equations
7.3.2 Approximate exponential map
Exercises to Chapter 7
8 Approximate symmetries
8.1 Definition of approximate symmetries
8.2 Calculation of approximate symmetries
8.2.1 Determining equations
8.2.2 Stable symmetries
8.2.3 Algorithm for calculation
8.3 Examples .
8.3.1 First example
8.3.2 Approximate commutator and Lie algebras
8.3.3 Second example
8.3.4 Third example
Exercises to Chapter 8
9 Applications
9.1 Integration of equations with a small parameter using approximate symmetries
9.1.1 Equation having no exact point symmetries
9.1.2 Utilization of stable symmetries
9.2 Approximately invariant solutions
9.2.1 Nonlinear wave equation
9.2.2 Approximate travelling waves of KdV equation
9.3 Approximate conservation laws
Exercises to Chapter 9
Assignment to Part II
Bibliography
Index
