Applications of analytic and geometric methods to nonlinear differential equations
By: edited by Peter A. Clarkson
Contributors: North Atlantic Treaty Organization. Scientific Affairs Division; NATO Advanced Research Workshop on Applications of Analytic and Geometric Methods to Nonlinear Differential Equations (1992 : Exeter, England)
Series: NATO ASI series. Series C, Mathematical and physical sciences -- no. 413; NATO ASI series -- no. 413.
Publisher: Spring
Published: 1993
Place of publication: Dordrecht
Language: eng
Format: [electronic resource] /
Number of pages: 477
ISBN-13: 9789401049245; 9789401120821
Subjects: Nonlinear Differential equations; Mathematical physics; Congresses; Numerical solutions; Solitons; Twistor theory
Description: In the study of integrable systems, two different approaches in particular have attracted considerable attention during the past twenty years. (1) The inverse scattering transform (IST), using complex function theory, which has been employed to solve many physically significant equations, the `soliton' equations. (2) Twistor theory, using differential geometry, which has been used to solve the self-dual Yang--Mills (SDYM) equations, a four-dimensional system having important applications in mathematical physics. Both soliton and the SDYM equations have rich algebraic structures which have been extensively studied. Recently, it has been conjectured that, in some sense, all soliton equations arise as special cases of the SDYM equations; subsequently many have been discovered as either exact or asymptotic reductions of the SDYM equations.

Consequently what seems to be emerging is that a natural, physically significant system such as the SDYM equations provides the basis for a unifying framework underlying this class of integrable systems, i.e. `soliton' systems. This book contains several articles on the reduction of the SDYM equations to soliton equations and the relationship between the IST and twistor methods. The majority of nonlinear evolution equations are nonintegrable, and so asymptotic, numerical perturbation and reduction techniques are often used to study such equations. This book also contains articles on perturbed soliton equations. Painlevé analysis of partial differential equations, studies of the Painlevé equations and symmetry reductions of nonlinear partial differential equations.

(ABSTRACT) In the study of integrable systems, two different approaches in particular have attracted considerable attention during the past twenty years; the inverse scattering transform (IST), for `soliton' equations and twistor theory, for the self-dual Yang--Mills (SDYM) equations. This book contains several articles on the reduction of the SDYM equations to soliton equations and the relationship between the IST and twistor methods. Additionally, it contains articles on perturbed soliton equations, Painlevé analysis of partial differential equations, studies of the Painlevé equations and symmetry reductions of nonlinear partial differential equations.
Notes: Softcover reprint of the hardcover 1st edition originally published by Kluwer in 1993.

"Proceedings of the NATO Advanced Research Workshop on Applications of Analytic and Geometric Methods to Nonlinear Differential Equations, Exeter, U.K., July 14-19, 1992"--T.p. verso.

"Published in cooperation with NATO Scientific Affairs Division."

Includes bibliographical references and indexes.