Analytic-Bilinear Approach to Integrable Hierarchies / Edition 1

by L.V. Bogdanov

Overview

The subject of this book is the hierarchies of integrable equations connected with the one-component and multi component loop groups. There are many publications on this subject, and it is rather well defined. Thus, the author would like t.o explain why he has taken the risk of revisiting the subject. The Sato Grassmannian approach, and other approaches standard in this context, reveal deep mathematical structures in the base of the integrable hi­ erarchies. These approaches concentrate mostly on the algebraic picture, and they use a language suitable for applications to quantum field theory. Another well-known approach, the a-dressing method, developed by S. V. Manakov and V.E. Zakharov, is oriented mostly to particular systems and ex­ act classes of their solutions. There is more emphasis on analytic properties, and the technique is connected with standard complex analysis. The language of the a-dressing method is suitable for applications to integrable nonlinear PDEs, integrable nonlinear discrete equations, and, as recently discovered, for t.he applications of integrable systems to continuous and discret.e geometry. The primary motivation of the author was to formalize the approach to int.e­ grable hierarchies that was developed in the context of the a-dressing method, preserving the analytic struetures characteristic for this method, but omitting the peculiarit.ies of the construetive scheme. And it was desirable to find a start.­

Product Details

ISBN-13:	9780792359197
Publisher:	Springer Netherlands
Publication date:	08/31/1999
Series:	Mathematics and Its Applications , #493
Edition description:	1999
Pages:	267
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

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.