Approximate and Renormgroup Symmetries deals with approximate transformation groups, symmetries of integro-differential equations and renormgroup symmetries. It includes a concise and self-contained introduction to basic concepts and methods of Lie group analysis, and provides an easy-to-follow introduction to the theory of approximate transformation groups and symmetries of integrodifferential equations.

The book is designed for specialists in nonlinear physics--mathematicians and non-mathematicians--interested in methods of applied group analysis for investigating nonlinear problems in physical science and engineering.

1 Lie Group Analysis in Outline

 1.1 Continuous point transformation groups

1.1.1 One-parameter groups

1.1.2 Infinitesimal transformations

1.1.3 Lie equations

1.1.4 Exponential map

1.1,5 Canonical variables

1.1,6 Invariants and invariant equations

 1.2 Symmetries of ordinary differential equations

1.2.1 Frame of differential equations

1.2.2 Extension of group actions to derivatives

1.2.3 Generators of prolonged groups

1.2.4 Definition of a symmetry group

1.2.5 Main property of symmetry groups

1.2.6 Calculation of infinitesimal symmetries

1.2.7 An example

1.2.8 Lie algebras

 1.3 Integration of first-order equations

1.3.1 Lie's integrating factor

1.3.2 Method of canonical variables

 1.4 Integration of second-order equations

1.4.1 Canonical variables in Lie algebras L2

1.4.2 Integration method

 1.5 Symmetries of partial differential equations

1.5.1 Main concepts illustrated by evolution equations

1.5.2 Invariant solutions

1.5.3 Group transformations of solutions

 1.6 Three definitions of symmetry groups

1.6.1 Frame and extended frame

1.6.2 First definition of symmetry group

1.6.3 Second definition

1.6.4 Third definition

 1.7 Lie-Backlund transformation groups

1.7.1 Lie-Backlund operators

1.7.2 Lie-Backlund equations and their integration

1.7.3 Lie-Backlund symmetries

 References

2 Approximate Transformation Groups and Symmetries

 2.1 Approximate transformation groups

2.1.1 Notation and definitions

2.1.2 Approximate Lie equations

2.1.3 Approximate exponential map

 2.2 Approximate symmetries

2.2.1 Definition of approximate symmetries

2.2.2 Determining equations & Stable symmetries

2.2.3 Calculation of approximate symmetries

2.2.4 Examples of approximate symmetries

2.2.5 Integration using approximate symmetries

2.2.6 Integration using stable symmetries

2.2.7 Approximately invariant solutions

2.2.8 Approximate conservation laws (first integrals)

 References

3 Symmetries of Integro-Differential Equations

 3.1 Definition and infinitesimal test

3.1.1 Definition of symmetry group

3.1.2 Variational derivative for functionals

3.1.3 Infinitesimal criterion

3.1.4 Prolongation on nonlocal variables

 3.2 Calculation of symmetries illustrated by Vlasov-Maxwell equations

3.2.1 One-dimensional electron gas

3.2.2 Three-dimensional plasma kinetic equations

3.2.3 Plasma kinetic equations with Lagrangian velocity

3.2.4 Electron-ion plasma equations in quasi-neutral approximation

 References

4 Renormgroup Symmetries

 4.1 Introduction

 4.2 Renormgroup algorithm

4.2.1 Basic manifold

4.2.2 Admitted group

4.2.3 Restriction of admitted group on solutions

4.2.4 Renormgroup invariant solutions

 4.3 Examples

4.3.1 Modified Burgers equation

4.3.2 Example from geometrical optics

4.3.3 Method based on embedding equations

4.3.4 Renormgroup and differential constraints

 References

5 Applications of Renormgroup Symmetries

 5.1 Nonlinear optics

5.1.1 Nonlinear geometrical optics

5.1.2 Nonlinear wave optics

5.1.3 Renormgroup algorithm using functionals

 5.2 Plasma physics

5.2.1 Harmonics generation in inhomogeneous plasma

5.2.2 Nonlinear dielectric permittivity of plasma

5.2.3 Adiabatic expansion of plasma bunches

 References

Index