By C. Rogers;W. K. Schief
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.
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Additional info for Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory
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  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  and Sym et al. in . 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.
Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory by C. Rogers;W. K. Schief