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基本説明
Presents elementary twistor theory as a universal technique for solving differential equations in applied mathematics and theoretical physics.
Full 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.
Contents
Preface ; 1. Integrability in classical mechanics ; 2. Soliton equations and the Inverse Scattering Transform ; 3. The hamiltonian formalism and the zero-curvature representation ; 4. Lie symmetries and reductions ; 5. The Lagrangian formalism and field theory ; 6. Gauge field theory ; 7. Integrability of ASDYM and twistor theory ; 8. Symmetry reductions and the integrable chiral model ; 9. Gravitational instantons ; 10. Anti-self-dual conformal structures ; Appendix A: Manifolds and Topology ; Appendix B: Complex analysis ; Appendix C: Overdetermined PDEs ; Index
