Download Bäcklund and Darboux Transformations: Geometry and Modern by C. Rogers;W. K. Schief PDF

By C. Rogers;W. K. Schief

This publication describes the extraordinary connections that exist among the classical differential geometry of surfaces and smooth soliton thought. The authors additionally discover the huge physique of literature from the 19th and early 20th centuries by means of such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on alterations of privileged periods of surfaces which depart key geometric homes unchanged. well-liked among those are Bäcklund-Darboux differences with their awesome linked nonlinear superposition rules and significance in soliton thought.

Show description

Read or Download Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory PDF

Similar geometry books

Laplacian on Riemannian manifold

This article on research on Riemannian manifolds is an intensive advent to subject matters coated in complex examine monographs on Atiyah-Singer index concept. the most subject matter is the research of warmth circulation linked to the Laplacians on differential types. this offers a unified therapy of Hodge thought and the supersymmetric evidence of the Chern-Gauss-Bonnet theorem.

Geometry of Sporadic Groups II: Representations and Amalgams (Encyclopedia of Mathematics and its Applications 91)

This moment quantity in a two-volume set presents a whole self-contained facts of the type of geometries linked to sporadic easy teams: Petersen and tilde geometries. It features a examine of the representations of the geometries into consideration in GF(2)-vector areas in addition to in a few non-Abelian teams.

Geometric Computations with Interval and New Robust Methods: Applications in Computer Graphics, GIS and Computational Geometry

This undergraduate and postgraduate textual content will familiarise readers with period mathematics and similar instruments to achieve trustworthy and verified effects and logically right judgements for quite a few geometric computations, and the potential for relieving the consequences of the blunders. It additionally considers computations on geometric point-sets, that are neither strong nor trustworthy in processing with commonplace tools.

Extra info for Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory

Example text

It was in 1978 that Harrison first derived a B¨acklund transformation for the Ernst equation. Independently, in 1979, Neugebauer constructed another B¨acklund transformation which subsequently has been shown to be a basic building block for all other B¨acklund transformations admitted by the Ernst equation. 6 opens with a description of the seminal Neugebauer transformation couched in terms of pseudopotentials. It is shown that it incorporates both the Ehlers and Matzner-Misner transformations.

3 deals tend to soliton surfaces linked to the AKNS class r = −q. with the iteration of elementary matrix Darboux transformations and a pivotal 14 General Introduction and Outline commutativity property is established. In geometric terms, it is shown that repetition of matrix Darboux transformations generates a suite of surfaces whose neighbouring members possess the constant length property. To conclude, iterated matrix Darboux transformations are exploited to construct a permutability theorem generic to the AKNS class r = −q¯ of soliton equations.

N . Thus, at each application of the B¨acklund transformation, a new B¨acklund parameter ␤i is introduced and an ith order soliton generated. 3. 2 Physical Applications Seeger et al. [345] exploited the permutability theorem for the sine-Gordon equation to investigate interaction properties of kink and breather-type solutions in connection with a crystal dislocation model. 4 Pseudospherical Soliton Surfaces. 3. A Bianchi lattice. of the propagation of ultrashort optical pulses in a resonant medium.

Download PDF sample

Rated 4.61 of 5 – based on 15 votes