Basic methods of soliton theory / Ivan Cherednik.

Format:

Book

Author/Creator:

Cherednik, Ivan.

Series:

Advanced series in mathematical physics ; v. 25.

Advanced series in mathematical physics ; v. 25

Language:

English

Subjects (All):

Solitons--Mathematics.

Solitons.

Differential equations, Nonlinear--Numerical solutions.

Differential equations, Nonlinear.

Geometry, Algebraic.

Physical Description:

1 online resource (264 p.)

Place of Publication:

Singapore ; River Edge, N.J. : World Scientific, c1996.

Language Note:

English

Summary:

In the 25 years of its existence Soliton Theory has drastically expanded our understanding of "integrability" and contributed a lot to the reunification of Mathematics and Physics in the range from deep algebraic geometry and modern representation theory to quantum field theory and optical transmission lines.The book is a systematic introduction to the Soliton Theory with an emphasis on its background and algebraic aspects. It is the first one devoted to the general matrix soliton equations, which are of great importance for the foundations and the applications.Differential algebra (local cons

Contents:

Contents; Preface; Introduction; An outline of the main directions; 0.1. Plan of this book; 0.2. Chiral fields and Sin-Gordon equation; 0.3. Generalized Heisenberg magnet and VNS equation; 0.4. Four key constructions; 0.5. Basic notations; I. CONSERVATION LAWS & ALGEBRAIC-GEOMETRIC SOLUTIONS; 1. Local conservation laws; 1.1. Formal Jost functions; 1.2. Basic constructions; 1.3. Formal jets; 1.4. Direct calculations (Riccati equation); 1.5. Local laws modulo exact derivatives; 1.6. Comments; 2. Generalized Lax equations; 2.1. Equations of the GHM and the VNS type

2.2. Jost functions as operators2.3. Abstract fractional powers (generating operator); 2.4. Generalized Lax equations (relations to VNS); 2.5. Scalar operators (T-functions); 2.6. Comments; 3. Algebraic-geometric solutions of basic equations; 3.1. Baker functions; 3.2. The main construction; 3.3. Reality conditions; 3.4. Curves with an involution; 3.5. Application: discrete PCF equation; 3.6. Comments; 4. Algebraic-geometric solutions of Sin-Gordon NS etc.; 4.1. Sin-Gordon equation S2-fields; 4.2. VNS equation; 4.3. Relations with the constructions in 3; 4.4. Application: the duality equation

4.5. CommentsII BACKLUND TRANSFORMS AND INVERSE PROBLEM; 1. Backlund transformations; 1.1. Transformations of basic equations; 1.2. Backlund transformations of Un On and Sn-1-fields; 1.3. Sin-Gordon equation; 1.4. Application: local conservation laws for the Sin-Gordon equation and Sn-1-fields; 1.5. Darboux transformation; Nonlinear Schrodinger equation; 1.6. Comments; 2. Introduction to the scattering theory; 2.1. Monodromy matrix; 2.2. Analytic continuations; 2.3. Variants; 2.4. Riemann-Hilbert problem; 2.5. Variational derivatives of entries of T (Poisson brackets); 2.6. Comments

3. Applications of the inverse problem method3.1. Inverse problem for the basic equations; 3.2. Asymptotic expansions and trace formulae; 3.3. Examples of reduction (On Sn-1-fields; n = 2); 3.4. Scattering data for certain solutions of the NS equation; 3.5. Application: DNS equation; 3.6. Comments; References; Index

Notes:

Description based upon print version of record.

Includes bibliographical references (p. 239-248) and index.

ISBN:

9789812798220

9812798226