Download PDF by C. Rogers, W. K. Schief: Backlund & Darboux Transformations
By C. Rogers, W. K. Schief
This booklet describes the impressive connections that exist among the classical differential geometry of surfaces and glossy soliton thought. The authors additionally discover the vast physique of literature from the 19th and early 20th centuries by way of such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on variations of privileged periods of surfaces which depart key geometric houses unchanged. well known among those are Bäcklund-Darboux ameliorations with their striking linked nonlinear superposition ideas and value in soliton thought.
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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 particular, an helicoid generated by the tractrix is pseudospherical and is known as a Dini surface. In terms of the parameters x, y as above, its coordinate vector 4 A change in the value of the constant of integration ␣ is merely equivalent to a change in the origin.
Accordingly, the vector r − r joining corresponding points on and is tangential to . It is recalled that it is tangential to by construction. 43) L cos( − ) C, 2 whence r u · Nu = 0, r u · Nv = r v · Nu = − The 2nd fundamental form for 1 sin(2 − ), r v · Nv = 0. 40) shows that is a pseudospherical surface parametrised by arc length along asymptotic lines. 45) where plays the same role in relation to as is played by in relation to . In particular, must satisfy the sine-Gordon equation uv = 1 sin .
54) where ␣ is an arbitrary constant of integration. 55) which have the characteristic hump shape associated with a soliton. Remarkably, analytic expressions for multi-soliton solutions which encapsulate their nonlinear interaction may now be obtained by an entirely algebraic procedure. This is a consequence of an elegant nonlinear superposition principle derived from the auto-B¨acklund transformation B␤ and originally set down by Bianchi  in 1892. 25) and that 1 and 2 are the B¨acklund transforms of via B␤1 and B␤2 , that is, 1 = B␤1 (), 2 = B␤2 ().
Backlund & Darboux Transformations by C. Rogers, W. K. Schief