Approximate and Renormgroup Symmetries
Description
PDF pp. 86ff. are on the Maxwell-Vlasov equation.
The book is dedicated to new trends in applied group analysis, focused on the computation of approximate symmetries of differential equation systems containing a small parameter. The methods of renormalization group (or "renormgroup'') theory are put in the context of approximate symmetry transformations. In addition, integro-differential equations are also discussed. The convenience of canonical group variables leaving invariant integration variables is stressed.
The contents are as follows: 1. Lie group analysis in outline (a short review on Lie symmetries); 2. Approximate transformation groups and symmetries; 3. Symmetries of integro-differential equations; 4. Renormgroup symmetries; 5. Applications of renormgroup symmetries.
The notion of renormgroup comes from quantum field theory, where the renormalization group was based upon finite (Dyson) transformations. Given a system of differential equations, the renormgroup method starts with the inclusion of all parameters, both from the equations and from the boundary conditions on which a particular solution depends, in group transformations. Hence the small parameter(s) are also transformed. The next steps of the renormgroup algorithm comprise the calculation of the admitted group, the restriction of admitted group on solutions and the search for an analytic invariant solution. Moreover, the calculation of approximate conservation laws can also be considered.
Many worked out examples are offered illustrating the general theory. In particular some basic plasma physics models are treated in detail, namely the Vlasov-Poisson and Vlasov-Maxwell systems, both in relativistic and non-relativistic conditions. Other plasma examples are found, for instance, on harmonic generation in inhomogeneous plasma and the adiabatic expansion of plasma bunches. This is a short, self-contained book providing a valuable reference for the application-oriented reader. Reviewed by Fernando Haas
Introduction
"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 integro-differential 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.
Dr. N.H. Ibragimov is a professor at the Department of Mathematics and Science, Research Centre ALGA, Sweden. He is widely regarded as one of the world's foremost experts in the field of symmetry analysis of differential equations; Dr. V. F. Kovalev is a leading scientist at the Institute for Mathematical Modeling, Russian Academy of Science, Moscow.
Front Matter
Pages i-iii
Lie Group Analysis in Outline
Pages 1-44
Approximate Transformation Groups and Symmetries
Pages 45-72
Symmetries of Integro-Differential Equations
Pages 73-94
Renormgroup Symmetries
Pages 95-111
Applications of Renormgroup Symmetries
Pages 113-141
Back Matter
Pages 143-145