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.

**Read Online or Download Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory PDF**

**Best geometry books**

**An Algebraic Approach to Geometry: Geometric Trilogy II by Francis Borceux PDF**

This can be a unified therapy of many of the algebraic methods to geometric areas. The examine of algebraic curves within the advanced projective aircraft is the average hyperlink among linear geometry at an undergraduate point and algebraic geometry at a graduate point, and it's also a major subject in geometric functions, corresponding to cryptography.

**Get Geometry of Cauchy-Riemann Submanifolds PDF**

This booklet gathers contributions by means of revered specialists at the idea of isometric immersions among Riemannian manifolds, and specializes in the geometry of CR buildings on submanifolds in Hermitian manifolds. CR buildings are a package deal theoretic recast of the tangential Cauchy–Riemann equations in advanced research concerning numerous advanced variables.

**Additional info for Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory**

**Sample text**

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.