Ebook: 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 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.

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Content:
Front Matter....Pages i-x
SDYM Hierarchies and Classical Soliton Systems....Pages 1-8
Twistor Theory, Self-Duality and Integrability....Pages 9-16
Twistor Theory and the Schlesinger Equations....Pages 17-25
Soliton Equations and Connections with Self-Dual Curvature....Pages 27-39
Smooth Static Solutions of the Einstein/Yang-Mills Equations....Pages 41-45
Extended Structures in (2 + 1) Dimensions....Pages 47-63
Null Reductions of the Yang-Mills Self-Duality Equations and Integrable Models in (2 + l)-Dimensions....Pages 65-75
Continuous and Discrete SDYM, and Reductions....Pages 77-80
Rapidly Forced Burgers Equation....Pages 81-87
Nonlinear Evolution Equations from an Inverse Spectral Problem....Pages 89-108
Universal Integrable Nonlinear PDEs....Pages 109-114
Construction of Reflectionless Potentials with Infinitely Many Discrete Eigenvalues....Pages 115-123
Coupling of Completely Integrable Systems: The Perturbation Bundle....Pages 125-138
The Negative Weight KP Hierarchy....Pages 139-148
The Complete Solution to the Constant Quantum Yang-Baxter Equation in Two Dimensions....Pages 149-154
Stimulated Raman Scattering: An Integrable System Which Blows up in Finite Time....Pages 155-161
Integrabile Quantum Mappings and Quantization Aspects of Integrable Discrete-Time Systems....Pages 163-182
Darboux Transformations in (2 + 1)-Dimensions....Pages 183-192
Darboux Theorems and the KP Hierarchy....Pages 193-206
Localized Soliton Solutions for the Davey-Stewartson I and Davey-Stewartson III Equations....Pages 207-215
On Gasdynamic-Solitonic Connections....Pages 217-228
Integrable Cellular Automata and Integrable Algebraic and Functional Equations....Pages 229-240
Classical Differential Geometry and Integrability of Systems of Hydrodynamic Type....Pages 241-249
On the Elliptic 2 + 1 Toda Equation....Pages 251-256
Canonical Bilinear Systems and Soliton Resonances....Pages 257-270
A Perturbative Extension of the Painlev? Test....Pages 271-280
Exact Solutions to the Complex Ginzburg-Landau Equation from a Linear System....Pages 281-286
Generalised Solutions of the Perturbed KdV Equation for Convecting Fluids....Pages 287-288
Modified Singular Manifold Expansion: Application to the Boussinesq and Mikhailov-Shabat Systems....Pages 289-298
Discrete Painlev? Equations: Derivation and Properties....Pages 299-313
A Study of the Fourth Painlev? Equation....Pages 315-330
A Local Asymptotic Method of Seeing the Natural Barrier of the Solutions of the Chazy Equation....Pages 331-339
Rational Solutions and B?cklund Transformations for the Third Painlev? Equation....Pages 341-352
Some Extensions of the Truncation Process in Painlev? Analysis....Pages 353-362
Potential Symmetries and Linearization....Pages 363-373
Nonclassical Symmetry Reductions and Exact Solutions of Nonlinear Reaction-Diffusion Equations....Pages 375-389
Boundary Conditions on Similarity Solutions....Pages 391-393
Symmetry Reductions and Exact Solutions of the Davey-Stewartson System....Pages 395-404
Dimensional Reduction for Equations Involving Discrete and Continuous Variables....Pages 405-413
Symmetry Reductions and Exact Solutions for a Generalised Boussinesq Equation....Pages 415-430
Continuous Symmetries and Painlev? Reduction of the Kac-Van Moerbeke Equation....Pages 431-439
A New Approach to the Search for Analytical Solutions of Second order Autonomous Nonlinear Differential Equations....Pages 441-453
Back Matter....Pages 455-477

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Content:
Front Matter....Pages i-x
SDYM Hierarchies and Classical Soliton Systems....Pages 1-8
Twistor Theory, Self-Duality and Integrability....Pages 9-16
Twistor Theory and the Schlesinger Equations....Pages 17-25
Soliton Equations and Connections with Self-Dual Curvature....Pages 27-39
Smooth Static Solutions of the Einstein/Yang-Mills Equations....Pages 41-45
Extended Structures in (2 + 1) Dimensions....Pages 47-63
Null Reductions of the Yang-Mills Self-Duality Equations and Integrable Models in (2 + l)-Dimensions....Pages 65-75
Continuous and Discrete SDYM, and Reductions....Pages 77-80
Rapidly Forced Burgers Equation....Pages 81-87
Nonlinear Evolution Equations from an Inverse Spectral Problem....Pages 89-108
Universal Integrable Nonlinear PDEs....Pages 109-114
Construction of Reflectionless Potentials with Infinitely Many Discrete Eigenvalues....Pages 115-123
Coupling of Completely Integrable Systems: The Perturbation Bundle....Pages 125-138
The Negative Weight KP Hierarchy....Pages 139-148
The Complete Solution to the Constant Quantum Yang-Baxter Equation in Two Dimensions....Pages 149-154
Stimulated Raman Scattering: An Integrable System Which Blows up in Finite Time....Pages 155-161
Integrabile Quantum Mappings and Quantization Aspects of Integrable Discrete-Time Systems....Pages 163-182
Darboux Transformations in (2 + 1)-Dimensions....Pages 183-192
Darboux Theorems and the KP Hierarchy....Pages 193-206
Localized Soliton Solutions for the Davey-Stewartson I and Davey-Stewartson III Equations....Pages 207-215
On Gasdynamic-Solitonic Connections....Pages 217-228
Integrable Cellular Automata and Integrable Algebraic and Functional Equations....Pages 229-240
Classical Differential Geometry and Integrability of Systems of Hydrodynamic Type....Pages 241-249
On the Elliptic 2 + 1 Toda Equation....Pages 251-256
Canonical Bilinear Systems and Soliton Resonances....Pages 257-270
A Perturbative Extension of the Painlev? Test....Pages 271-280
Exact Solutions to the Complex Ginzburg-Landau Equation from a Linear System....Pages 281-286
Generalised Solutions of the Perturbed KdV Equation for Convecting Fluids....Pages 287-288
Modified Singular Manifold Expansion: Application to the Boussinesq and Mikhailov-Shabat Systems....Pages 289-298
Discrete Painlev? Equations: Derivation and Properties....Pages 299-313
A Study of the Fourth Painlev? Equation....Pages 315-330
A Local Asymptotic Method of Seeing the Natural Barrier of the Solutions of the Chazy Equation....Pages 331-339
Rational Solutions and B?cklund Transformations for the Third Painlev? Equation....Pages 341-352
Some Extensions of the Truncation Process in Painlev? Analysis....Pages 353-362
Potential Symmetries and Linearization....Pages 363-373
Nonclassical Symmetry Reductions and Exact Solutions of Nonlinear Reaction-Diffusion Equations....Pages 375-389
Boundary Conditions on Similarity Solutions....Pages 391-393
Symmetry Reductions and Exact Solutions of the Davey-Stewartson System....Pages 395-404
Dimensional Reduction for Equations Involving Discrete and Continuous Variables....Pages 405-413
Symmetry Reductions and Exact Solutions for a Generalised Boussinesq Equation....Pages 415-430
Continuous Symmetries and Painlev? Reduction of the Kac-Van Moerbeke Equation....Pages 431-439
A New Approach to the Search for Analytical Solutions of Second order Autonomous Nonlinear Differential Equations....Pages 441-453
Back Matter....Pages 455-477
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