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By C. Rogers, W. K. Schief

This booklet describes the extraordinary connections that exist among the classical differential geometry of surfaces and glossy soliton conception. The authors additionally discover the broad physique of literature from the 19th and early 20th centuries by way of such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on alterations of privileged periods of surfaces which go away key geometric homes unchanged. favorite among those are Bäcklund-Darboux adjustments with their extraordinary linked nonlinear superposition rules and value in soliton conception.

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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. The composition laws for Neugebauer transformations are then set down, and the mechanism for their iteration is explained.

130) where The metric of the spherical representation may be expressed in terms of the fundamental forms of . 131) for a hyperbolic surface with Gaussian curvature K=− 1 .

The preceding provides an historical motivation, both with regard to theory and application, for beginning our study of B¨acklund transformations with the classical result for the sine-Gordon equation. It will be seen that this B¨acklund transformation, in fact, corresponds to a conjugation of invariant transformations due to Bianchi and Lie. The Lie symmetry serves to intrude a key B¨acklund parameter into the Bianchi transformation which enables its iteration and the generation thereby of what are, in physical terms, multi-soliton solutions.

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