Approximate and renormgroup symmetries
Starting with Sophus Lie's original approach of using symmetries to solve differential equations the authors present a very compact, self-contained introduction to the new field in modern group analysis of approximate transformation groups and especially renormalization group symmetries. Many applications are given for nonlinear problems from all parts of mathematical physics. This survey is based on the many books and papers the authors published on this subject. Table of Contents: Ch. 1 Lie Group Analysis in Outline 1.1 Continuous point transformation groups. 1.2 Symmetries of ordinary differential equations. 1.3 Integration of first-order equations. 1.4 Integration of second-order equations. 1.5 Symmetries of partial differential equations. 1.6 Three definitions of symmetry groups. 1.7 Lie-Bäcklund transformation groups. Ch. 2 Approximate Transformation Groups and Symmetries 2.1 Approximate transformation groups: Notation and definitions, Approximate Lie equations, Approximate exponential map; 2.2 Approximate symmetries, Definition of approximate symmetries, Determining equations and Stable symmetries, Calculation of approximate symmetries, Examples of approximate symmetries, Integration using approximate symmetries, Integration using stable symmetries, Approximately invariant solutions, Approximate conservation laws (first integrals). Ch. 3 Symmetries of Integro-Differential Equations 3.1 Definition and infinitesimal test: Definition of symmetry group, Variational derivative for functionals, Infinitesimal criterion, Prolongation on non local variables; 3.2 Calculation of symmetries illustrated by Vlasov-Maxwell equations: One-dimensional electron gas, Three-dimensional plasma kinetic equations, Plasma kinetic equations with Lagrangian velocity, Electron-ion plasma equations in quasi-neutral approximation. Ch. 4 Renormgroup Symmetries 4.1 Introduction; 4.2 Renormgroup algorithm: Basic manifold, Admitted group, Restriction of admitted group on solutions, Renormgroup invariant solutions; 4.3 Examples: Modified Burgers equation, Example from geometrical optics, Method based on embedding equations, Renormgroup and differential constraints). Ch. 5 Applications of Renormgroup Symmetries 5.1 Nonlinear optics: Nonlinear geometrical optics liS, Nonlinear wave optics, Renormgroup algorithm using functionals); 5.2 Plasma physics: Harmonics generation in inhomogeneous plasma, Nonlinear dielectric permittivity of plasma, Adiabatic expansion of plasma bunches. References. Index
scientific article; zbMATH DE number 5528185