Integrable Systems: Twistors, Loop Groups, and Riemann Surfaces
N. J. Hitchin, Nigel Hitchin, Savilian Professor of Geometry N J Hitchin, G. B. Segal, R. S. Ward, Lowndean Professor of Geometry and Astronomy G B Segal
Clarendon Press, Mar 18, 1999 - Mathematics - 136 pages
This textbook is designed to give graduate students an understanding of integrable systems via the study of Riemann surfaces, loop groups, and twistors. The book has its origins in a series of lecture courses given by the authors, all of whom are internationally known mathematicians and renowned expositors. It is written in an accessible and informal style, and fills a gap in the existing literature. The introduction by Nigel Hitchin addresses the meaning of integrability: how do we recognize an integrable system? His own contribution then develops connections with algebraic geometry, and includes an introduction to Riemann surfaces, sheaves, and line bundles. Graeme Segal takes the Kortewegde Vries and nonlinear Schrödinger equations as central examples, and explores the mathematical structures underlying the inverse scattering transform. He explains the roles of loop groups, the Grassmannian, and algebraic curves. In the final part of the book, Richard Ward explores the connection between integrability and the self-dual Yang-Mills equations, and describes the correspondence between solutions to integrable equations and holomorphic vector bundles over twistor space.
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List of contributors
Riemann surfaces and integrable systems
Some classical integrable systems
The nonlinear Schrödinger equation and
The KdV equation as an Euler equation
Local conservation laws
Integrable Systems and Twistors
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CGLnC coefficients cohomology groups commutative compact support complex conserved quantities constant coordinate curvature defined denote described det(w det(X dim H'(M dimension dimensional eigenfunction eigenvalues equa example fact finite flow geodesic given Grassmannian Grº Grres Hamiltonian harmonic maps holo holomorphic function holomorphic line bundle holomorphic map holomorphic sections holomorphic vector bundle holonomy Hom(Q integrable systems isomorphism Jacobian KdV equation kernel Lax form Lax pair Lax pair equations Lie algebra linear long exact sequence loop group matrix meromorphic function morphic multiplication non-linear open sets operator Lu parameter poles proof Proposition quotient rapidly decreasing real axis Riemann sphere Riemann surfaces rigid body Segal Serre duality sheaf short exact sequence solitons solutions spectral curve spectrum subspace symplectic manifold tangent theorem topology torus transition functions trivial bundle upper half-plane vanishes vector field vector space Yang–Mills equations zero