Solitons, Instantons, and Twistors

Paperback | January 17, 2010

byMaciej Dunajski

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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 importantroles 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 ofanti-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, andanti-self-duality equations. The three appendices cover basic differential geometry, complex manifold theory, and the exterior differential system.

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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 importantroles in pure and applied mathematics as well ...

Maciej Dunajski read physics in Lodz, Poland and received a PhD in mathematics from Oxford University where he held a Senior Scholarship at Merton College. After spending four years as a lecturer in the Mathematical Institute in Oxford where he was a member of Roger Penrose's research group, he moved to Cambridge, where he holds a Fel...
Format:PaperbackDimensions:368 pages, 9.21 × 6.14 × 0.01 inPublished:January 17, 2010Publisher:Oxford University PressLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:0198570635

ISBN - 13:9780198570639

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Table of Contents

Preface1. Integrability in classical mechanics2. Soliton equations and the Inverse Scattering Transform3. The hamiltonian formalism and the zero-curvature representation4. Lie symmetries and reductions5. The Lagrangian formalism and field theory6. Gauge field theory7. Integrability of ASDYM and twistor theory8. Symmetry reductions and the integrable chiral model9. Gravitational instantons10. Anti-self-dual conformal structuresAppendix A: Manifolds and TopologyAppendix B: Complex analysisAppendix C: Overdetermined PDEsIndex