Nonlinear Symmetries and Nonlinear Equations

The study of (nonlinear) dift"erential equations was S. Lie's motivation when he created what is now known as Lie groups and Lie algebras; nevertheless, although Lie group and algebra theory flourished and was applied to a number of dift"erent physical situations -up to the point that a lot, if not most, of current fun damental elementary particles physics is actually (physical interpretation of) group theory -the application of symmetry methods to dift"erential equations remained a sleeping beauty for many, many years. The main reason for this lies probably in a fact that is quite clear to any beginner in the field. Namely, the formidable comple:rity ofthe (algebraic, not numerical!) computations involved in Lie method. I think this does not account completely for this oblivion: in other fields of Physics very hard analytical computations have been worked through; anyway, one easily understands that systems of dOlens of coupled PDEs do not seem very attractive, nor a very practical computational tool.

Klappentext

This book serves as an introduction to the use of nonlinear symmetries in studying, simplifying and solving nonlinear equations. Part I provides a self-contained introduction to the theory. This emphasizes an intuitive understanding of jet spaces and the geometry of differential equations, and a special treatment of evolution problems and dynamical systems, including original results. In Part II the theory is applied to equivariant dynamics, to bifurcation theory and to gauge symmetries, reporting recent results by the author. In particular, the fundamental results of equivariant bifurcation theory are extended to the case of nonlinear symmetries. The final part of the book gives an overview of new developments, including a number of applications, mainly in the physical sciences. An extensive and up-to-date list of references dealing with nonlinear symmetries completes the volume. br/ This volume will be of interest to researchers in mathematics and mathematical physics. br/

Inhalt
I - Geometric setting.- a): Equations and functions as geometrical objects.- b): Symmetry.- References.- II - Symmetries and their use.- 1. Symmetry of a given equation.- 2. Linear and C-linearizable equations.- 3. Equations with a given symmetry.- 4. Canonical coordinates.- 5. Symmetry and reduction of algebraic equations.- 6. Symmetry and reduction of ODEs.- 7. Symmetry and symmetric solutions of PDEs.- 8. Conditional symmetries.- 9. Conditional symmetries and boundary conditions.- References.- III - Examples.- 1. Symmetry of algebraic equations.- 2. Symmetry of ODEs (one-soliton KdV).- 3. Symmetry of evolution PDEs (the heat equation).- 4. Table of prolongations for ODEs.- 5. Table of prolongations for PDEs.- IV - Evolution equations.- a): Evolution equations - general features.- b): Dynamical systems (ODEs).- c): Periodic solutions.- d): Evolution PDEs.- References.- V - Variational problems.- 1. Variational symmetries and variational problems.- 2. Variational symmetries and conservation laws: Lagrangian mechanics and Noether theorem.- 3. Conserved quantities for higher order variational problems: the general Noether theorem.- 4. Noether theorem and divergence symmetries.- 5. Variational symmetries and reduction of order.- 6. Variational symmetries, conservation laws, and the Noether theorem for infinite dimensional variational problems.- References.- VI - Bifurcation problems.- 1. Bifurcation problems: general setting.- 2. Bifurcation theory and linear symmetry.- 3. Lie-point symmetries and bifurcation.- 4. Symmetries of systems of ODEs depending on a parameter.- 5. Bifurcation points and symmetry algebra.- 6. Extensions.- References.- VII - Gauge theories.- 1. Symmetry breaking in potential problems and gauge theories.- 2. Strata in RN.- 3. Michel's theorem.- 4.Zero-th order gauge functionals.- 5. Discussion.- 6. First order gauge functionals.- 7. Geometry and stratification of ?.- 8. Stratification of gauge orbit space.- 9. Maximal strata in gauge orbit space.- 10. The equivariant branching lemma.- 11. A reduction lemma for gauge invariant potentials.- 12. Some examples of reduction.- 13. Base space symmetries.- 14. A scenario for pattern formation.- 15. A scenario for phase coexistence.- References.- VIII - Reduction and equivariant branching lemma.- 1. General setting (ODEs).- 2. The reduction lemma.- 3. The equivariant branching lemma.- 4. General setting (PDEs).- 5. Gauge symmetries and Lie point vector fields.- 6. Reduction lemma for gauge theories.- 7. Symmetric critical sections of gauge functionals.- 8. Equivariant branching lemma for gauge functionals.- 9. Evolution PDEs.- 10. Symmetries of evolution PDEs.- 11. Reduction lemma for evolution PDEs.- References.- IX - Further developements.- 1. Missing sections.- 2. Non Linear Superposition Principles.- 3. Symmetry and integrability - second order ODEs.- 4. Infinite dimensional (and Kac-Moody) Lie-point symmetry algebras.- 5. Symmetry classification of ODEs.- 6. The Lie determinant.- 7. Systems of linear second order ODEs.- 8. Cohomology and symmetry of differential equations.- 9. Contact symmetries of evolution equations.- 10. Conditional symmetries, and Boussinesq equation.- 11. Lie point symmetries and maps.- References.- X - Equations of Physics.- 1. Fokker-Planck type equations.- 2. Schroedinger equation for atoms and molecules.- 3. Einstein (vacuum) field equations.- 4. Landau-Ginzburg equation.- 5. The ?6 field theory (three dimensional Landau-Ginzburg equation).- 6. An equation arising in plasma physics.- 7. Navier-Stokes equations.- 8. Yang-Mills equations.-9. Lattice equations and the Toda lattice.- References.- References and bibliography.