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Analytic-Bilinear Approach to Integrable Hierarchies

Author: L V Bogdanov
Publisher: Dordrecht : Springer Netherlands : Imprint : Springer, 1999.
Series: Mathematics and Its Applications ;, 493.
Edition/Format:   eBook : Bibliographic data : EnglishView all editions and formats
Database:WorldCat
Summary:
This book presents the analytic-bilinear approach to integrable hierarchies, which gives a consistent and technically simple description of integrable hierarchies, and shows an easy and direct 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  Read more...
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Details

Genre/Form: Electronic books
Additional Physical Format: Printed edition:
Material Type: Bibliographic data, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: L V Bogdanov
ISBN: 9789401144957 9401144958 9789401059220 9401059225
OCLC Number: 840310307
Description: 1 online resource (276 pages).
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.
Series Title: Mathematics and Its Applications ;, 493.
Responsibility: by L.V. Bogdanov.

Abstract:

This book presents the analytic-bilinear approach to integrable hierarchies, which gives a consistent and technically simple description of integrable hierarchies, and shows an easy and direct 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 tau-function and its additional formulae. <br/> Audience: 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.

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