Applications of Analytic and Geometric Methods to Nonlinear Differential Equations

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 severalarticles 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.

Inhalt
I. Self-Dual Yang-Mills Equations.- SDYM Hierarchies and classical soliton systems.- Twistor theory, self-duality and integrability.- Twistor theory and the Schlesinger equations.- Soliton equations and connections with self-dual curvature.- Smooth static solutions of the Einstein/Yang-Mills equations.- Extended structures in (2 + 1) dimensions.- Null reductions of the Yang-Mills self-duality equations and integrable models in (2 + 1)-dimensions.- Continuous and discrete SDYM, and reductions.- II. Completely Integrable Equations.- Rapidly forced Burgers equation.- Nonlinear evolution equations from an inverse spectral problem.- Universal integrable nonlinear PDEs.- Construction of reflectionless potentials with infinitely many discrete eigenvalues.- Coupling of completely integrable systems: the perturbation bundle.- The negative weight KP hierarchy.- The complete solution to the constant quantum Yang-Baxter equation in two dimensions.- Stimulated Raman scattering: an integrable system which blows up in finite time.- Integrable quantum mappings and quantization aspects of integrable discrete-time systems.- Darboux transformations in (2 + 1)-dimensions.- Darboux theorems and the KP hierarchy.- Localized soliton solutions for the Davey-Stewartson I and Davey-Stewartson III equations.- On Gasdynamic-solitonic connections.- Integrable cellular automata and integrable algebraic and functional equations.- Classical differential geometry and integrability of systems of hydrodynamic type.- On the elliptic 2 + 1 Toda equation.- Canonical bilinear systems and soliton resonances.- III. Painlevé Equations and Painlevé Analysis.- A perturbative extension of the Painlevé test.- Exact solutions to the complex Ginzburg-Landau equation from a linear system.- Generalized solutions ofa perturbed KdV equation for convecting fluids.- Modified singular manifold expansion: application to the Boussinesq and Mikhailov-Shabat systems.- Discrete Painlevé equations: derivation and properties.- A study of the fourth Painlevé equation.- A local asymptotic method of seeing the natural barrier of the solutions of the Chazy equation.- Rational solutions and Bäcklund transformations for the third Painlevé equation.- Some extensions of the truncation process in Painlevé analysis.- IV. Symmetries of Differential Equations.- Potential symmetries and linearization.- Nonclassical symmetry reductions and exact solutions of nonlinear reaction-diffusion equations.- Boundary conditions on similarity solutions.- Symmetry reductions and exact solutions of the Davey-Stewartson system.- Dimensional reduction for equations involving discrete and continuous variables.- Symmetry reductions and exact solutions for a generalized Boussinesq equation.- Continuous symmetries and Painlevé reduction of the Kac-van Moerbeke equation.- A new approach to the search for analytical solutions of second order autonomous nonlinear differential equations.- Participants.- Contributors.- Contributor Index.- Author Index.