By C. Rogers;W. K. Schief
This publication describes the awesome connections that exist among the classical differential geometry of surfaces and sleek soliton thought. The authors additionally discover the broad physique of literature from the 19th and early 20th centuries through such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on variations of privileged sessions of surfaces which go away key geometric homes unchanged. renowned among those are Bäcklund-Darboux adjustments with their impressive linked nonlinear superposition ideas and value in soliton thought.
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Additional resources for Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory
The above result may be re-formulated. 136) by virtue of K=− f2 f2 1 = − = − . 137) Now, the Gauss-Weingarten equations imply that N satisfies the hyperbolic equation Nuv + 1 v 1 u Nu + Nv + FN = 0. 133) written in the form ru = Nu × N, rv = − Nv × N. 140) The latter relations are commonly referred to as the Lelieuvre formulae (see ).
Beltrami  subsequently gave the term pseudospherical to these surfaces and made important connections with Lobachevski’s non-Euclidean geometry. It was Bour , in 1862, who seems to have first set down what is now termed the sine-Gordon equation arising out of the compatibility conditions for the Gauss equations for pseudospherical surfaces expressed in asymptotic coordinates. In 1879, Bianchi  in his habilitation thesis presented, in mathematical terms, a geometric construction for pseudospherical surfaces.
This result was extended by B¨acklund  in 1883 to incorporate a key parameter which allows the iterative construction of such pseudospherical surfaces. The B¨acklund transformation was subsequently shown by Bianchi , in 1885, to be associated with an elegant invariance of the sine-Gordon equation. This invariance has become known as the B¨acklund transformation for the sine-Gordon equation. It includes an earlier parameter-independent result of Darboux . The B¨acklund transformation has important applications in soliton theory.