By C. Rogers;W. K. Schief
This booklet describes the striking connections that exist among the classical differential geometry of surfaces and sleek soliton thought. The authors additionally discover the wide physique of literature from the 19th and early 20th centuries via such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on alterations of privileged periods of surfaces which depart key geometric houses unchanged. sought after among those are Bäcklund-Darboux differences with their amazing linked nonlinear superposition ideas and value in soliton thought.
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Additional info for Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory
To conclude, an analogue of Bianchi’s classical transformation for pseudospherical surfaces is set down. In Chapter 7, the important connection between B¨acklund transformations and matrix Darboux transformations is established. 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.
101) is said to be parallel to . Here, c represents the constant distance along the normal between and ˜ . 104) while where ⑀ = ±1, depending on whether (1 − c1 )(1 − c2 ) is positive or negative. Moreover, e˜ = ⑀(1 − c1 )e, 5 f˜ = 0, g˜ = ⑀(1 − c2 )g. 105) In terms of the principal curvatures 1 , 2 the Gaussian curvature K and mean curvature M are 2 given respectively by K = 1 2 , M = 1 + 2 . 42 1 The Classical B¨acklund Transformation The parallel surface ˜ is seen to be parametrised along lines of curvature which correspond to those on the original surface .
It is recalled that a single application of the Harrison transformation produces the Schwarzschild solution on the Papapetrou background, while N applications with seed solution the Kerr black hole metric leads to a nonlinear superposition of N Kerr-NUT fields. 7, a matrix Darboux transformation for a generalised Ernst system and its specialisations to the Ernst equation and its dual are presented. 8, successive application of two such transformations is shown to lead to permutability theorems for the Ernst equation and its dual.
Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory by C. Rogers;W. K. Schief