nach oben

2007 | Buch

Kapitel lesen Erstes Kapitel lesen

Hamiltonian Methods in the Theory of Solitons

verfasst von: Ludwig D. Faddeev, Leon A. Takhtajan

Verlag: Springer Berlin Heidelberg

Buchreihe : Classics in Mathematics

Enthalten in: Professional Book Archive

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Einloggen, um Zugang zu erhalten

Über dieses Buch

This book presents the foundations of the inverse scattering method and its applications to the theory of solitons in such a form as we understand it in Leningrad. The concept of solitonwas introduced by Kruskal and Zabusky in 1965. A soliton (a solitary wave) is a localized particle-like solution of a nonlinear equation which describes excitations of finite energy and exhibits several characteristic features: propagation does not destroy the profile of a solitary wave; the interaction of several solitary waves amounts to their elastic scat tering, so that their total number and shape are preserved. Occasionally, the concept of the soliton is treated in a more general sense as a localized solu tion of finite energy. At present this concept is widely spread due to its universality and the abundance of applications in the analysis of various processes in nonlinear media. The inverse scattering method which is the mathematical basis of soliton theory has developed into a powerful tool of mathematical physics for studying nonlinear partial differential equations, almost as vigoraus as the Fourier transform. The book is based on the Hamiltonian interpretation of the method, hence the title. Methods of differential geometry and Hamiltonian formal ism in particular are very popular in modern mathematical physics. It is precisely the general Hamiltonian formalism that presents the inverse scat tering method in its most elegant form. Moreover, the Hamiltonian formal ism provides a link between classical and quantum mechanics.

Inhaltsverzeichnis

Frontmatter

Introduction

Abstract

Over the past fifteen years the theory of solitons and the related theory of integrable nonlinear evolution equations in two space-time dimensions has attracted a large number of research workers of different orientations ranging from algebraic geometry to applied hydrodynamics. Modern mathematical physics has witnessed the development of a vast new area of research devoted to this theory and called the inverse scattering method of solving nonlinear equations (other names are: the inverse spectral transform, the method of isospectral deformations and, more colloquially, the L-A pair method).

Ludwig D. Faddeev, Leon A. Takhtajan

The Nonlinear Schrödinger Equation (NS Model)

Frontmatter

Chapter I. Zero Curvature Representation

Abstract

The dynamical system to be considered is generated by the nonlinear equation

(1.1)

with the initial condition

(1.2)

Ludwig D. Faddeev, Leon A. Takhtajan

Chapter II. The Riemann Problem

Abstract

In Chapter I we analyzed the mapping

from the functions Ψ(x), to the transition coefficients and discrete spectrum of the auxiliary linear problem. We saw that for both rapidly decreasing and finite density boundary conditions this “change of variables” makes the dynamics quite simple because the time evolution of the transition coefficients for the continuous and discrete spectra becomes linear.

Ludwig D. Faddeev, Leon A. Takhtajan

Chapter III. The Hamiltonian Formulation

Abstract

In this chapter we return to the Hamiltonian formulation of the NS model in order to discuss the basic transformation of the inverse scattering method

from the Hamiltonian standpoint. We shall describe the Poisson structure on the scattering data of the auxiliary linear problem induced through f from the initial Poisson structure defined in Chapter I. Under the rapidly decreasing or finite density boundary conditions, the NS model proves to be a completely integrable system, with f defining a transformation to action-angle variables. In particular, we will show that the integrals of the motion introduced in Chapter I are in involution. In these terms scattering of solitons amounts to a simple canonical transformation.

Ludwig D. Faddeev, Leon A. Takhtajan

General Theory of Integrable Evolution Equations

Frontmatter

Chapter I. Basic Examples and Their General Properties

Abstract

In this chapter we shall give a list of typical examples and establish their general properties: the zero curvature representation and the Hamiltonian formulation. Then, motivated by these examples, we shall outline a general scheme for constructing integrable equations and their solutions based on the matrix Riemann problem. A detailed study of the most important models and the Hamiltonian interpretation of the general scheme will be presented in the following chapters. The examples to be considered fall into two classes: dynamical systems generated by partial differential evolution equations (continuous models), and evolution systems of difference type (lattice models).

Ludwig D. Faddeev, Leon A. Takhtajan

Chapter II. Fundamental Continuous Models

Abstract

We shall give a complete list of results pertaining to two fundamental continuous models, the HM and SG models. For the rapidly decreasing boundary conditions we shall analyze the mapping F from the initial data of the auxiliary linear problem to the transition coefficients and the discrete spectrum, and show how to solve the inverse problem, i. e. how to construct the mapping F⁻¹. We shall see that these models allow an r-matrix approach, which will enable us to show that F is a canonical transformation to variables of action-angle type. It will thus be proved that the HM and SG models are completely integrable Hamiltonian systems. We shall also present a Hamiltonian interpretation of the change to light-cone coordinates in the SG model. To conclude this chapter, we shall explain that in some sense the LL model is the most universal integrable system with two-dimensional auxiliary space.

Ludwig D. Faddeev, Leon A. Takhtajan

Chapter III. Fundamental Models on the Lattice

Abstract

Here we shall give a complete list of results pertaining to the Toda model, a fundamental model on the lattice. We will show that the r-matrix approach applies to this case and may be used to prove the complete integrability of the model in the quasi-periodic case. For the rapidly decreasing boundary conditions we will analyze the mapping F from the initial data of the auxiliary linear problem to the transition coefficients and outline a method for solving the inverse problem, i.e. for F⁻¹ constructing. On the basis of the r-matrix approach it will be shown that F is a canonical trans formation to action-angle type variables establishing the complete integrability of the Toda model in the rapidly decreasing case. We shall also define a lattice version of the LL model, the most general integrable lattice system with two-dimensional auxiliary space.

Ludwig D. Faddeev, Leon A. Takhtajan

Chapter IV. Lie-Algebraic Approach to the Classification and Analysis of Integrable Models

Abstract

In this chapter we shall summarize and generalize our experience in describing integrable models gained from the study of particular examples. The principal entities of the inverse scattering method and its Hamiltonian interpretation were the auxiliary linear problem operator L = d/dx − U(x, λ) and the fundamental Poisson brackets for U(x, λ) involving the r-matrix. Similar objects were introduced for lattice models. We will show that these notions have a simple geometric interpretation.

Ludwig D. Faddeev, Leon A. Takhtajan

Conclusion

Abstract

This conclusion is intended for those who have read the book to the end. We hope that the main text and the notes to separate chapters have furnished a convincing evidence of how rich from the mathematical point of view, both conceptually and technically, is the subject of solitons and integrable partial differential equations. In fact, the inverse scattering method naturally intertwines various branches of mathematics: differential geometry, the theory of Lie groups and Lie algebras and their representations, complex and functional analysis. All of them serve one common purpose, to classify integrable equations and describe their solutions. As a result, the traditional parts of these branches, such as Hamiltonian formalism, affine Lie algebras, or the Riemann problem are seen in a new light.

Ludwig D. Faddeev, Leon A. Takhtajan

Backmatter

Titel: Hamiltonian Methods in the Theory of Solitons
verfasst von: Ludwig D. Faddeev
Leon A. Takhtajan
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-69969-9
Print ISBN: 978-3-540-69843-2
DOI: https://doi.org/10.1007/978-3-540-69969-9