Overview

This book presents the analytic-bilinear approach to integrable hierarchies, which gives a consistent description of integrable hierarchies, and shows a straightforward way to understand rather complicated structures, using mostly standard complex analysis. The language of the analytic-bilinear approach is suitable for applications to integrable nonlinear PDEs, integrable nonlinear discrete equations, and for the applications of integrable systems to continuous and discrete geometry. This approach allows the representation of generalised hierarchies of integrable equations in a condensed form of finite functional equations, incorporating basic hierarchy, modified hierarchy, singularity manifold equation hierarchy and corresponding linear problems, which arise both in the compact discrete form and in the form of nonlinear partial differential equations. Different levels of generalised hierarchy are connected via invariants of Combescure symmetry transformation. The resolution of functional equations also leads to the &tgr;-function and its additional formulae. This book will be of interest to students and specialists whose work involves the theory of integrable systems, mathematical physics, topological groups, Lie groups, finite differences, functional equations, partial differential equations and functions of a complex variable.

Full Product Details

Author:   L.V. Bogdanov
Publisher:   Springer
Imprint:   Springer
Edition:   1999 ed.
Volume:   493
Dimensions:   Width: 15.50cm , Height: 1.70cm , Length: 23.50cm
Weight:   1.270kg
ISBN:  

9780792359197

ISBN 10:   0792359194
Pages:   267
Publication Date:   31 August 1999
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1 Introduction.- 1.1 Hirota Bilinear Identity.- 1.2 Meromorphic Loops, Determinant Formula and the ?-Function.- 1.3 Integrable Discrete Equations.- 1.4 From Discrete Equations to the Continuous Hierarchy.- 2 Hirota Bilinear Identity for the Cauchy Kernel.- 2.1 Boundary Problems for the ¯?-Operator in the Unit Disc.- 2.2 General Boundary Problems with Zero Index.- 2.3 Rational Deformations of the Boundary Problems.- 2.4 Hirota Bilinear Identity.- 2.5 Determinant Formula for Action of Meromorphic Loops on the Cauchy Kernel.- 2.6 ?-Function for the One-Component Case.- 3 Rational Loops and Integrable Discrete Equations. I: Zero Local Indices.- 3.1 One-Component Case.- 3.2 General Matrix Equations for the Multicomponent Case.- 4 Rational Loops and Integrable Discrete Equations. II: Two-Component Case.- 4.1 DS case.- 4.2 2DTL Case.- 5 Rational Loops and Integrable Discrete Equations. III: The General Case.- 5.1 General Multicomponent Case.- 5.2 ?-Function for the Multicomponent Case.- 5.3 Three-Component Case.- 5.4 Four-Component Case.- 5.5 Five-Component and Six-Component Cases.- 6 Generalized KP Hierarchy.- 6.1 Generalized Hirota Identity from the ¯?-Dressing Method.- 6.2 The Generalized KP Hierarchy.- 6.3 KP Hierarchy in the ‘Moving Frame’. Darboux Equations as the Horizontal Subhierarchy.- 6.4 Combescure Symmetry Transformations.- 6.5 ?-Function and Addition Formulae.- 6.6 ?-Function as a Functional.- 6.7 From the Discrete Case to the Continuous.- 7 Multicomponent Kp Hierarchy.- 7.1 Multicomponent Case with Zero Local Indices.- 7.2 ?-Function and Closed 1-Form for ?+N.- 7.3 Generalized DS Hierarchy.- 7.4 Loop Group ? and 2DTL Hierarchy.- 8 On The ¯?-Dressing Method.- 8.1 General Scheme.- 8.2 Matrix Lattice and q-Difference Darboux Equations.- 8.3 Special Cases of Nonlocal ¯?-Problem.- 8.4 On Some Equations, Integrable Via ¯?-Dressing Method.- 8.5 Solutions with Special Properties.- 8.6 Boussinesq Equation.- 8.7 Relativistically-Invariant Systems.- 8.8 Inverse Problems for the Differential Operator of Arbitrary Order on the Line.

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