Solitons, Instantons, and Twistors

Inbunden, Engelska, 2024

AvMaciej Dunajski,University of Cambridge) Dunajski, Maciej (Professor of Mathematical Physics, Professor of Mathematical Physics, Department of Applied Mathematics and Theoretical Physics

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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 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, vortices, gravitational instantons, twistor transforms, and anti-self-duality equations. The three appendices cover basic differential geometry, complex manifold theory, and the exterior differential system.

Produktinformation

Utgivningsdatum2024-05-20
Mått160 x 240 x 10 mm
Vikt812 g
FormatInbunden
SpråkEngelska
SerieOxford Graduate Texts in Mathematics
Antal sidor416
Upplaga2
FörlagOUP OXFORD
ISBN9780198872535

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Maciej Dunajski is a Fellow of Clare College, and a Professor of Mathematical Physics at the Department of Applied Mathematics and Theoretical Physics, at the University of Cambridge. His research interests are in differential and projective Geometry, Solitons, and General Theory of Relativity. In 2021 he was awarded the Atiyah Fellowship by the London Mathematical Society. Dunajski is the winner of the 2023 Gravity Research Foundation Award, and the author of Geometry: A Very Short Introduction (OUP 2022).

1: Integrability in classical mechanics2: Soliton equations and the inverse scattering transform3: Hamiltonian formalism and zero-curvature representation4: Lie symmetries and reductions5: Lagrangian formalism and field theory6: Gauge field theory7: Integrability of ASDYM and twistor theory8: Symmetry reductions and the integrable chiral model9: Vortices10: Gravitational instantons11: Anti-self-dual conformal structures

The great strength of this volume is how self-contained it is in its approach, one seldom needs to look elsewhere before delving headfirst into this volume. In my own work, this text is often my first go-to choice whenever I require reference.A particular strength of the text is its firm modern introduction to integrability, as well as its gentle and instructive introduction to gravitational instantons and twistor theory. This text is suitable for a graduate-level reader base whose education has at least included a first course in quantum field theory and general relativity.

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