Bäcklund and Darboux Transformations
Bäcklund and Darboux Transformations

Author: C. Rogers

Publisher: Cambridge University Press

Published: 2002-06-24

Total Pages: 436

ISBN-13: 9780521012881

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This book explores the deep and fascinating connections that exist between a ubiquitous class of physically important waves known as solitons and the theory of transformations of a privileged class of surfaces as they were studied by eminent geometers of the nineteenth century. Thus, nonlinear equations governing soliton propagation and also mathematical descriptions of their remarkable interaction properties are shown to arise naturally out of the classical differential geometry of surfaces and what are termed Bäcklund-Darboux transformations.This text, the first of its kind, is written in a straightforward manner and is punctuated by exercises to test the understanding of the reader. It is suitable for use in higher undergraduate or graduate level courses directed at applied mathematicians or mathematical physics.

Darboux Transformations and Solitons
Darboux Transformations and Solitons

Author: Vladimir B. Matveev

Publisher: Springer

Published: 1992-09-30

Total Pages: 122

ISBN-13: 9783662009246

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The modem theory of solitons was born in 1967 when Gardner, Greene, Kruskal and Miura related the solution of the Cauchy initial value problem for the Korteweg-de Vries equation to the inverse scattering problem for a one dimensional linear Schrödinger equation. Soliton theory is now a large part of theoretical and mathematical physics. An important method used to solve related equations is based on the Inverse Scattering Transform (IST). This IST method has been extended and applied to a large variety of (analytically) solvable non linear evolution equations, including many important examples describing phe nomena in nonlinear optics, solid state physics, hydrodynamics, theory of general relativity, plasma physics, etc. In the about twenty years of development the necessary mathematical tools have become rather sophisticated. They include the methods of algebraic geome try, the machinery of group representations, the theory of the local and nonlocal Riemann-Hilbert problem and many other "higher" levels of contemporary math ematics.

Darboux Transformations in Integrable Systems
Darboux Transformations in Integrable Systems

Author: Chaohao Gu

Publisher: Springer Science & Business Media

Published: 2006-07-09

Total Pages: 317

ISBN-13: 1402030886

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The Darboux transformation approach is one of the most effective methods for constructing explicit solutions of partial differential equations which are called integrable systems and play important roles in mechanics, physics and differential geometry. This book presents the Darboux transformations in matrix form and provides purely algebraic algorithms for constructing the explicit solutions. A basis for using symbolic computations to obtain the explicit exact solutions for many integrable systems is established. Moreover, the behavior of simple and multi-solutions, even in multi-dimensional cases, can be elucidated clearly. The method covers a series of important equations such as various kinds of AKNS systems in R1+n, harmonic maps from 2-dimensional manifolds, self-dual Yang-Mills fields and the generalizations to higher dimensional case, theory of line congruences in three dimensions or higher dimensional space etc. All these cases are explained in detail. This book contains many results that were obtained by the authors in the past few years. Audience: The book has been written for specialists, teachers and graduate students (or undergraduate students of higher grade) in mathematics and physics.

Soliton Theory and Its Applications
Soliton Theory and Its Applications

Author: Chaohao Gu

Publisher: Springer Science & Business Media

Published: 2013-03-14

Total Pages: 414

ISBN-13: 3662031027

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Soliton theory is an important branch of applied mathematics and mathematical physics. An active and productive field of research, it has important applications in fluid mechanics, nonlinear optics, classical and quantum fields theories etc. This book presents a broad view of soliton theory. It gives an expository survey of the most basic ideas and methods, such as physical background, inverse scattering, Backlünd transformations, finite-dimensional completely integrable systems, symmetry, Kac-moody algebra, solitons and differential geometry, numerical analysis for nonlinear waves, and gravitational solitons. Besides the essential points of the theory, several applications are sketched and some recent developments, partly by the authors and their collaborators, are presented.

Bäcklund and Darboux Transformations
Bäcklund and Darboux Transformations

Author: A. A. Coley

Publisher: American Mathematical Soc.

Published: 2001-01-01

Total Pages: 460

ISBN-13: 9780821870259

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This book is devoted to a classical topic that has undergone rapid and fruitful development over the past 25 years, namely Backlund and Darboux transformations and their applications in the theory of integrable systems, also known as soliton theory. The book consists of two parts. The first is a series of introductory pedagogical lectures presented by leading experts in the field. They are devoted respectively to Backlund transformations of Painleve equations, to the dressing methodand Backlund and Darboux transformations, and to the classical geometry of Backlund transformations and their applications to soliton theory. The second part contains original contributions that represent new developments in the theory and applications of these transformations. Both the introductorylectures and the original talks were presented at an International Workshop that took place in Halifax, Nova Scotia (Canada). This volume covers virtually all recent developments in the theory and applications of Backlund and Darboux transformations.

Nonlinear Waves
Nonlinear Waves

Author: Lokenath Debnath

Publisher: CUP Archive

Published: 1983-12-30

Total Pages: 376

ISBN-13: 9780521254687

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The outcome of a conference held in East Carolina University in June 1982, this book provides an account of developments in the theory and application of nonlinear waves in both fluids and plasmas. Twenty-two contributors from eight countries here cover all the main fields of research, including nonlinear water waves, K-dV equations, solitions and inverse scattering transforms, stability of solitary waves, resonant wave interactions, nonlinear evolution equations, nonlinear wave phenomena in plasmas, recurrence phenomena in nonlinear wave systems, and the structure and dynamics of envelope solitions in plasmas.

Discrete and Continuous Nonlinear Schrödinger Systems
Discrete and Continuous Nonlinear Schrödinger Systems

Author: M. J. Ablowitz

Publisher: Cambridge University Press

Published: 2004

Total Pages: 276

ISBN-13: 9780521534376

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This book presents a detailed mathematical analysis of scattering theory, obtains soliton solutions, and analyzes soliton interactions, both scalar and vector.

The Direct Method in Soliton Theory
The Direct Method in Soliton Theory

Author: Ryogo Hirota

Publisher: Cambridge University Press

Published: 2004-07-22

Total Pages: 220

ISBN-13: 9780521836609

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Account of method of solving soliton equations by the inventor of the method.

Nonlinear Dispersive Waves
Nonlinear Dispersive Waves

Author: Mark J. Ablowitz

Publisher: Cambridge University Press

Published: 2011-09-08

Total Pages: 363

ISBN-13: 1139503480

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The field of nonlinear dispersive waves has developed enormously since the work of Stokes, Boussinesq and Korteweg–de Vries (KdV) in the nineteenth century. In the 1960s, researchers developed effective asymptotic methods for deriving nonlinear wave equations, such as the KdV equation, governing a broad class of physical phenomena that admit special solutions including those commonly known as solitons. This book describes the underlying approximation techniques and methods for finding solutions to these and other equations. The concepts and methods covered include wave dispersion, asymptotic analysis, perturbation theory, the method of multiple scales, deep and shallow water waves, nonlinear optics including fiber optic communications, mode-locked lasers and dispersion-managed wave phenomena. Most chapters feature exercise sets, making the book suitable for advanced courses or for self-directed learning. Graduate students and researchers will find this an excellent entry to a thriving area at the intersection of applied mathematics, engineering and physical science.