By C. Rogers, W. K. Schief
This publication describes the striking connections that exist among the classical differential geometry of surfaces and sleek soliton thought. The authors additionally discover the large physique of literature from the 19th and early 20th centuries by way of such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on modifications of privileged periods of surfaces which go away key geometric homes unchanged. admired among those are Bäcklund-Darboux adjustments with their awesome linked nonlinear superposition ideas and value in soliton conception.
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Extra info for Backlund & Darboux Transformations
A geometric interpretation follows for a Miura-type transformation which links the NLS hierarchy and this associated eigenfunction hierarchy. 20) XT = ± 1 + X 2Z ZZ which is linked to the mKdV equation by a combination of reciprocal and gauge transformations. The presence of loop solitons in the complex NLS hierarchy is then established. Loop solitons are seen to be naturally associated with the generation of soliton surfaces. 21) and, more generally, of the Dym hierarchy t = −1 (−D 3r I r )n x , n = 1, 2, .
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. 2, the requirement of invariance of the projective Gauss-Mainardi-Codazzi equations under a simple Lie point symmetry is shown to lead to a specialisation which may be identiﬁed as the Euler-Lagrange equations associated with projective-minimal surfaces.
37) now yields − 2 + 2 = u v + ␤ sin 2 − 1 sin = . 46). 48) that is, the tangent planes at corresponding points on and meet at a constant angle where ␤ = tan( /2). 49) so that these tangent planes are orthogonal. B¨acklund’s relaxation of the orthogonality requirement allows the key parameter ␤ to be inserted into the Bianchi transformation. In fact, the B¨acklund transformation B␤ may be viewed as a composition of a Bianchi transformation with a simple Lie group invariance. 47).