Solitons, Instantons, and Twistors: (19 Oxford Graduate Texts in Mathematics)

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.

Table of 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

About the Author :
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 holds a Fellowship and lectureship at Clare College and a Newton Trust Lectureship at the Department of Applied Mathematics and Theoretical Physics. Dunajski specialises in twistor theory and differential geometric approaches to integrability and solitons. He is married with two sons.

Review :
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. As an introduction to an exciting area of research, this book is excellent because it is not only accessible but self contained. A wonderful feature of the book is the clear and informative explanation of the topics and the wealth of examples. The presentation style of the book means that it is accessible to readers ranging from advanced undergraduates doing research to experts. It would be an excellent textbook for a course at the advanced undergraduate level or graduate level in either mathematics or physics. This book will become a standard on the subject. The typesetting of the book is very clean, with nicely sized fonts and clean uniform notation. It includes 35 illustrations which helpfully illustrated text. It is my pleasure to highly recommend it to anyone from an advanced undergraduate to a researcher in the fields covered. The author has done a remarkable job of weaving these topics together in an engaging and readable book...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.