By C. Rogers;W. K. Schief
This e-book describes the amazing connections that exist among the classical differential geometry of surfaces and glossy soliton concept. The authors additionally discover the wide physique of literature from the 19th and early 20th centuries by means of such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on changes of privileged sessions of surfaces which depart key geometric homes unchanged. well-known among those are Bäcklund-Darboux adjustments with their impressive linked nonlinear superposition rules and value in soliton thought.
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Extra resources for Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory
29), we have r u = [1 − L(u − u ) sin ]A + L(u − u ) cos B + r v = (cos − Lv sin )A + (sin + Lv cos )B + L sin C, L sin( − )C. 33) and 1 1∓ v = L L2 1 − 2 sin( − ). 37) be a pseudospherical surface parametrised by on the angle in order that arc length along asymptotic lines. 37), is sufficient in this regard. 25). 39) ␤ so that r u · r v = cos(2 − ) and the 1st fundamental form of I = du 2 + 2 cos(2 − ) dudv + dv 2 . 30), it is seen that (r − r) · N = 0. Accordingly, the vector r − r joining corresponding points on and is tangential to .
89) where L = sin . 90) where 0 is a seed solution, 1 = B␤1(0 ), 2 = B␤2(0 ) and 12 = B␤2(1 ) = B␤1(2 ). 70), the two-soliton solution 2 + 1 1 − 2 sin sinh 2 2 . 6. A two-soliton pseudospherical surface. 89). 6. 4 Breathers There exists an important subclass of entrapped periodic two-soliton solutions known as breathers. Here, an analytic expression for the breather solution is obtained via the permutability theorem, and associated pseudospherical surfaces are constructed.
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. The latter have been studied in detail by Steuerwald . To every rational number d between zero and unity, there corresponds a pseudospherical stationary breather surface which is periodic in the y-parameter.