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