By C. Rogers;W. K. Schief
This ebook describes the outstanding connections that exist among the classical differential geometry of surfaces and sleek soliton conception. The authors additionally discover the wide physique of literature from the 19th and early 20th centuries by way of such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on changes of privileged periods of surfaces which go away key geometric houses unchanged. renowned among those are Bäcklund-Darboux ameliorations with their impressive linked nonlinear superposition rules and value in soliton concept.
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Additional info for Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory
66), respectively, with = . 71). 4 Pseudospherical Soliton Surfaces. 4. The Beltrami pseudosphere ( = /2). In terms of the lines of curvature parameters x and y, the position vector of the pseudosphere is4 sech x y + ␣ cos y x + ␣ sin r(x, y) = sech x x + ␣ − tanh +␣ . 84) Here, the coordinate lines x = const and y = const are parallels and meridians, respectively. 4. 2 A Pseudospherical Helicoid The surface generated by a curve which is rotated about an axis and simultaneously translated parallel to that axis in such a way that the ratio of the velocity of translation to the velocity of rotation is constant, is known as an helicoid.
In fact, the B¨acklund transformation B␤ may be viewed as a composition of a Bianchi transformation with a simple Lie group invariance. 47). 29) and delivers a one-parameter family of pseudospherical surfaces associated with a given solution of the sine-Gordon equation. 25). Let denote the B¨acklund transform of via B␤ . 52) where L = sin . 26). 25) and leaves it invariant. 25) holds: otherwise the compatibility condition uv = vu is not satisfied. • B␤ contains a parameter ␤ = tan( /2) injected into the underlying Bianchi transformation by a Lie group invariance.
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. 4). Chapter 9 describes developments in soliton theory which are linked to the geometry of projective-minimal and isothermal-asymptotic surfaces. 1, the analogues of the Gauss-Weingarten and Gauss-Mainardi-Codazzi equations are set down for surfaces in projective space P3 , and certain projective invariants are recorded.