Backlund & Darboux Transformations by C. Rogers, W. K. Schief

By C. Rogers, W. K. Schief

This ebook describes the awesome connections that exist among the classical differential geometry of surfaces and sleek soliton conception. The authors additionally discover the broad physique of literature from the 19th and early 20th centuries by means of such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on variations of privileged sessions of surfaces which go away key geometric houses unchanged. favourite among those are Bäcklund-Darboux alterations with their notable linked nonlinear superposition rules and significance in soliton conception.

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Induced B¨acklund Transformation 43 then the Gauss equation reduces to an integrable elliptic sinh-Gordon equation. 113) which is independent of M if and only if c = ±␳ . 114) ˜ is constant and we can formulate the following theorem due to In this case, M Bonnet. Theorem 1. A surface of constant positive Gaussian curvature K = 1/␳2 admits two parallel surfaces of constant mean curvature ±1/(2␳ ). The distance to the surface of positive Gaussian curvature is ␳ . 110). 101), generate mutually orthogonal surfaces which are, in turn, orthogonal to the parallel surfaces ˜ .

26) induces the B¨acklund transformation B␤ operating at the nonlinear level. In that B␤ represents a correspondence between solutions of the same equation, it is commonly termed an auto-B¨acklund transformation. 25) may be generated by purely algebraic procedures. The algorithmic nature of the latter makes them well-suited to implementation by symbolic computation packages. Such nonlinear superposition principles are generically associated with the autoB¨acklund transformations admitted by solitonic equations.

This surface is known as the pseudosphere. 66). 81) whence I = ␳ 2 cot2 ␺ d␺ 2 + ␳ 2 sin2 ␺ d␩2 , II = ␳ cot ␺ d␺ 2 − ␳ sin ␺ cos ␺ d␩2 . 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.

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