By C. Rogers;W. K. Schief

ISBN-10: 052181331X

ISBN-13: 9780521813310

This publication describes the striking connections that exist among the classical differential geometry of surfaces and smooth soliton concept. The authors additionally discover the huge physique of literature from the 19th and early 20th centuries by means of such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on differences of privileged periods of surfaces which depart key geometric homes unchanged. popular among those are Bäcklund-Darboux adjustments with their extraordinary linked nonlinear superposition rules and significance in soliton concept.

Show description

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

Best geometry books

Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover Books on Mathematics)

Scholars and pros within the fields of arithmetic, physics, engineering, and economics will locate this reference paintings valuable. A vintage source for operating with distinctive capabilities, common trig, and exponential logarithmic definitions and extensions, it positive factors 29 units of tables, a few to as excessive as 20 areas.

Calculus: Early Transcendental Functions

Scholars who've used Smith/Minton's "Calculus" say it really is more straightforward to learn than the other math ebook they have used. Smith/Minton wrote the ebook for the scholars who will use it, in a language that they comprehend, and with the expectancy that their backgrounds could have gaps. Smith/Minton offer unheard of, reality-based functions that attract scholars' pursuits and show the beauty of math on the planet round us.

Effective Methods in Algebraic Geometry

The symposium "MEGA-90 - potent equipment in Algebraic Geome­ test" was once held in Castiglioncello (Livorno, Italy) in April 17-211990. the topics - we quote from the "Call for papers" - have been the fol­ lowing: - potent tools and complexity matters in commutative algebra, professional­ jective geometry, actual geometry, algebraic quantity thought - Algebraic geometric tools in algebraic computing Contributions in comparable fields (computational points of staff idea, differential algebra and geometry, algebraic and differential topology, and so forth.

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

Sample text

97) This is known as the stationary breather since it is not translated as y evolves. 98) . For stationary breather solutions, sin ␨1 = sin ␨2 = ␹2 = 1 , c 1 (cx − idy) = ␹¯ 1 , r2 = r¯1 , ␪2 = ␪¯1 . 100)   cos y cos(dy) 2 cosh(cx) 2d  sin y cos(dy) , + 2 2 c d cosh (cx) + c2 sin2 (dy) −sinh(cx) 40 1 The Classical B¨acklund Transformation √ where c = 1 − d 2 . It is readily verified that the lines of curvature y = const are planar and, accordingly, the above pseudospherical surfaces constitute Enneper surfaces.

Indeed, it appears that the property of invariance under B¨acklund and associated Darboux transformations as originated in [92] is enjoyed by all soliton equations. The contribution of Bianchi and Darboux to the geometry of surfaces and, in particular, the role of B¨acklund transformations preserving certain geometric properties have been discussed by Chern [77] and Sym et al. in [385]. It is with B¨acklund and Darboux transformations, their geometric origins and their application in modern soliton theory that we shall be concerned in the present monograph.

A generic geometric property of these B¨acklund transformations is established, namely that they preserve distance between corresponding points. 3 deals tend to soliton surfaces linked to the AKNS class r = −q. with the iteration of elementary matrix Darboux transformations and a pivotal 14 General Introduction and Outline commutativity property is established. In geometric terms, it is shown that repetition of matrix Darboux transformations generates a suite of surfaces whose neighbouring members possess the constant length property.

Download PDF sample

Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory by C. Rogers;W. K. Schief

by Christopher

Rated 4.36 of 5 – based on 37 votes