This title presents an introduction to the classical treatment of Backlund and general surface transformations; and includes detailed and accessible techniques for constructing both groups of tranformations which will be of great value to the scientist and engineer in the analysis of mathematical models of physical phenomena. Classical and recent examples of Backlund transformations as applied to geometry, nonlinear optics, turbulence models, nonlinear waves and quantum mechanics are given. The authors discuss applications of Lie-Backlund transformations in mechanics, quantum mechanics, gas dynamics, hydrodynamics, and relativity.
The authors discuss applications of Lie-Backlund transformations in mechanics, quantum mechanics, gas dynamics, hydrodynamics, and relativity.
Author: Robert L. Anderson
Category: Bäcklund transformations
Author: Robert Leonard Anderson
Author: Robert Leonard Anderson
The aim of this paper is to show that the theory of jet bundles supplies the appropriate setting for the study of Backlund trans formations. These transformations are used to solve certain partial differential equations, particularly non-linear evolution equations. Of course jets have been employed for some time in the theory of partial differential equations, but so far little use has been made of them in applications. In the meanwhile, substantial progress has been made in the study of non-linear evolution equations. This work has been encouraged by the dis covery of remarkable properties of some such equations, for example the existence of soliton solutions and of infinite se quences of conservation laws. Among the techniques devised to deal with these equations are the inverse scattering method and the Backlund transformation. In our opinion the jet bundle formulation offers a unifying geometrical framework for under standing the properties of non-linear evolution equations and the techniques used to deal with them, although we do not consider all of these properties and techniques here. The relevance of the theory of jet bundles lS that it legitimates the practice of regarding the partial derivatives of field variables as independent quantities. Since Backlund trans formations require from the outset manipulation of these partial derivatives, and repeated shifts of point of view about which variables are dependent on which, this geometrical setting clari fies and simplifies the concepts involved, and offers the prospect of bringing coherence to a variety of disparate results.
This work has been encouraged by the dis covery of remarkable properties of some such equations, for example the existence of soliton solutions and of infinite se quences of conservation laws.
Author: F.A.E. Pirani
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.
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 ...
Author: C. Rogers
Publisher: Cambridge University Press
Ba?cklund Transformations and Their Applications
transformation have important applications in mathematical physics and continuum mechanics and, accordingly, each will be treated in detail in this monograph. An extensive bibliography of the early literature on Bäcklund transformations ...
Publisher: Academic Press
Lie - Bäcklund Transformations in Applications by Robert L. Anderson and Nail H. Ibragimov This first title in the new series , SIAM Studies in Applied Mathematics , presents an introduction to the classical treatment of Bäcklund and ...
Author: Society for Industrial and Applied Mathematics
Category: Control theory
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.
R. L. Anderson and N. J. Ibragimov , Lie - Bäcklund transformations in applications , SIAM , Philadelphia , 1979 . 3. H.-H. Chen , General derivation of Bäcklund transformation from inverse scattering problems , Phys . Rev. Lett .
Author: A. A. Coley
Publisher: American Mathematical Soc.
Chapter three specifically aims to trace a few applications of Lie - Backlund transformations to differential equations with intent to hint at the potentialities of such transformations ; a Lie - Backlund operator generates a Lie ...
Category: Mechanics, Applied