Download Bäcklund and Darboux Transformations: Geometry and Modern by C. Rogers;W. K. Schief PDF

By C. Rogers;W. K. Schief

This ebook describes the extraordinary connections that exist among the classical differential geometry of surfaces and glossy soliton thought. The authors additionally discover the large physique of literature from the 19th and early 20th centuries via such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on differences of privileged sessions of surfaces which go away key geometric homes unchanged. in demand among those are Bäcklund-Darboux changes with their striking linked nonlinear superposition rules and significance in soliton concept.

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These surfaces generalise the constant mean curvature surfaces mentioned in the previous subsection. 6 The Bianchi System. 125) − ab sin ␻ = 0. 126) so that the Gaussian curvature takes the form K=− 1 < 0. 127) In 1890, Bianchi [33] presented a purely geometric construction of hyperbolic surfaces subject to the constraint ␳ uv = 0. 128) 46 1 The Classical B¨acklund Transformation These surfaces have been termed Bianchi surfaces. Here, we derive a B¨acklund transformation for hyperbolic surfaces without restriction as to the Gaussian curvature.

11) augmented by the ‘Theorema egregium’ of Gauss. 12) in terms of E, F, G alone according to, in Liouville’s representation, K= 1 H H E 2 11 v − H E . 13) u In physical terms, the ‘Theorema egregium’ implies that the total curvature of a surface is invariant under bending without stretching. 20 1 The Classical B¨acklund Transformation If the total curvature of is negative, that is, if is a hyperbolic surface, then the asymptotic lines on may be taken as parametric curves. 16) 2 12 = E G u − F Ev .

3. 2 Physical Applications Seeger et al. [345] 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. Therein, analytic expressions for ‘2N ␲’ light pulses were obtained via the nonlinear superposition principle. These 2N ␲ pulses exhibit the distinctive property that they ultimately decompose into N stable 2␲ pulses.

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