By Robert Hermann

**Read Online or Download Geometry of Non-Linear Differential Equations, Backlund Transformations, and Solitons, Part B PDF**

**Similar geometry books**

**An Algebraic Approach to Geometry: Geometric Trilogy II - download pdf or read online**

This can be a unified therapy of a few of the algebraic methods to geometric areas. The examine of algebraic curves within the complicated projective aircraft is the normal 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, resembling cryptography.

This ebook gathers contributions through revered specialists at the thought of isometric immersions among Riemannian manifolds, and makes a speciality of 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 complicated research regarding a number of advanced variables.

**Extra info for Geometry of Non-Linear Differential Equations, Backlund Transformations, and Solitons, Part B**

**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.