By C. Rogers, W. K. Schief
This e-book describes the impressive connections that exist among the classical differential geometry of surfaces and glossy soliton conception. The authors additionally discover the wide physique of literature from the 19th and early 20th centuries through such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on changes of privileged sessions of surfaces which go away key geometric homes unchanged. favourite among those are Bäcklund-Darboux changes with their awesome linked nonlinear superposition ideas and significance in soliton idea.
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Additional resources for Backlund & Darboux Transformations
In . It is with B¨acklund and Darboux transformations, their geometric origins and their application in modern soliton theory that we shall be concerned in the present monograph. 1 The Gauss-Weingarten Equations for Hyperbolic Surfaces. Pseudospherical Surfaces. The Sine-Gordon Equation Here, the study of pseudospherical surfaces is set in the broader context of hyperbolic surfaces via a nonlinear system due to Bianchi . The background is that of basic classical differential geometry of curves and surfaces to be found in such standard works as do Carmo  or Struick .
The importance in general relativity of the resultant so-called Geroch transformations is then discussed. 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.
1, the SymTafel formula for the generic position vector of soliton surfaces is applied to show that the original B¨acklund transformation for the construction of pseudospherical surfaces provides a prototype for a matrix version of a classical Darboux transformation. It is established that the B¨acklund transformation for the construction of NLS soliton surfaces can likewise be represented as a matrix Darboux transformation which acts on the underlying su(2) representation. 2, an elementary matrix Darboux transformation is constructed which leaves invariant the AKNS representation for the NLS hierarchy.