Solitons, Instantons, and Twistors
Series: Oxford Graduate Texts in Mathematics; 19;
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Product details:
- Publisher OUP Oxford
- Date of Publication 10 December 2009
- ISBN 9780198570622
- Binding Hardback
- No. of pages374 pages
- Size 242x163x25 mm
- Weight 705 g
- Language English
- Illustrations 35 illustrations 0
Categories
Short description:
A text aimed at third year undergraduates and graduates in mathematics and physics, presenting elementary twistor theory as a universal technique for solving differential equations in applied mathematics and theoretical physics.
MoreLong description:
Most nonlinear differential equations arising in natural sciences admit chaotic behaviour and cannot be solved analytically. Integrable systems lie on the other extreme. They possess regular, stable, and well behaved solutions known as solitons and instantons. These solutions play important roles in pure and applied mathematics as well as in theoretical physics where they describe configurations topologically different from vacuum. While integrable equations in lower space-time dimensions can be solved using the inverse scattering transform, the higher-dimensional examples of anti-self-dual Yang-Mills and Einstein equations require twistor theory. Both techniques rely on an ability to represent nonlinear equations as compatibility conditions for overdetermined systems of linear differential equations.
The book provides a self-contained and accessible introduction to the subject. It starts with an introduction to integrability of ordinary and partial differential equations. Subsequent chapters explore symmetry analysis, gauge theory, gravitational instantons, twistor transforms, and anti-self-duality equations. The three appendices cover basic differential geometry, complex manifold theory, and the exterior differential system.
My view is that the book is a success. I have no hesitation in recommending the book as a textbook/reference for advanced undergraduates (Mmath or other masters level), and for researchers as well. It is also very valuable as a crossover book: showing researchers in other disciplines how some of this new theory motivated by cosmology can be introduced into other areas such as fluid mechanics.
Table of Contents:
Preface
Integrability in classical mechanics
Soliton equations and the Inverse Scattering Transform
The hamiltonian formalism and the zero-curvature representation
Lie symmetries and reductions
The Lagrangian formalism and field theory
Gauge field theory
Integrability of ASDYM and twistor theory
Symmetry reductions and the integrable chiral model
Gravitational instantons
Anti-self-dual conformal structures
Appendix A: Manifolds and Topology
Appendix B: Complex analysis
Appendix C: Overdetermined PDEs
Index
