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。
This is an introduction to a new field in applied group analysis.Namely，the book deals with the so-called renormalization group（briefly renormgroup）symmetries considered in the framework of approximate transformation groups.The notion of the renormalization group and the renormalization group method were introduced in theoretical physics by N.N.Bogoliubov and D.V.Shirkov in 1950s.Renormgroup symmetries provide a basis for the renormgroup algorithm for improving solutions to boundary value problems by converting"less applicable solutions"into"more applicable solutions".The algorithm is particularly useful for improving approximate solutions given by the perturbation theory.
We present in a concise form the essence of the mathematical apparatus for computing approximate and renormgroup symmetries using the infinitesimal techniques of the modern group analysis.In order to make the book self-contained，we provide in Chapter 1 an outline of basic notions from the classical Lie group analysis of differential equations.Chapters 2 and 3 reflect new trends in the modern group analysis.Chapter 2 contains a brief discussion of approximate transformation groups.In Chapter 3 we discuss methods for calculating symmetries of integro-differential equations.Renormgroup symmetries are introduced and illustrated by several examples in Chapter 4.The renormgroup algorithm is applied to various nonlinear problems in mathematical physics in Chapter 5.
The authors wish to express their gratitude to Professor Dmitry V.Shirkov，a world leader in the study of renormalization groups in quantum field theory.Our collaboration with him over many years plays a decisive role in preparing the"physical part"（Chapters 3，4 and 5）of the monograph.We also would like to say a word of genuine appreciation in memory of late Dr.Veniamin V.Pustovalov who made our collaboration possible and who inspired many ideas that form a ground of this book.
Dr.Nail H.Ibragimov is a professor at the Department of Mathematics and Science，Research Centre ALGA，Sweden.He is widely regarded as one of the worlds foremost experts in the field of symmetry analysis of differential equations；Vladimir F.Kovalev is a leading scientist at the Institute for Mathematical Modeling，Russian Academy of Science，Moscow.
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 Lies 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
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