By C. Rogers;W. K. Schief
This publication describes the extraordinary connections that exist among the classical differential geometry of surfaces and glossy soliton conception. The authors additionally discover the wide physique of literature from the 19th and early 20th centuries through such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on variations of privileged sessions of surfaces which depart key geometric houses unchanged. renowned among those are Bäcklund-Darboux modifications with their notable linked nonlinear superposition rules and significance in soliton conception.
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Extra resources for Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory
93) over a range which depicts both the nonlinear interaction and eventual decomposition into two separate single soliton solutions. 5 Parallel Surfaces. 101) is said to be parallel to . Here, c represents the constant distance along the normal between and ˜ . 104) while where ⑀ = ±1, depending on whether (1 − c1 )(1 − c2 ) is positive or negative. Moreover, e˜ = ⑀(1 − c1 )e, 5 f˜ = 0, g˜ = ⑀(1 − c2 )g. 105) In terms of the principal curvatures 1 , 2 the Gaussian curvature K and mean curvature M are 2 given respectively by K = 1 2 , M = 1 + 2 .
110). 101), generate mutually orthogonal surfaces which are, in turn, orthogonal to the parallel surfaces ˜ . 2 An Induced B¨acklund Transformation If a B¨acklund transformation is known for a class of surfaces , then a B¨acklund transformation is naturally induced for the class of parallel surfaces. 111). 41) whence, in terms of curvature coordinates, N = L [−cos( − ) + cos( + )]rx sin 2 + = L L L␤ [−cos( − ) − cos( + )]r y + 1 − N sin 2 sin cos L␤ rx − ry + 1 − N. 118) sin + (c / ) cos cos + (c / ) sin r˜ x − r˜ y cos − (c / ) sin sin + (c / ) cos +⑀ c 1− L␤ ˜ − c N.
7. 7. Stationary breather pseudospherical surfaces : p q = 14 , 34 , 15 , 12 . 5 Parallel Surfaces. 89), one can obtain the position vector of the pseudospherical surfaces associated with N-soliton solutions of the sineGordon equation as generated via iteration of the permutability theorem. Exercises 1. 89). 2. 93) over a range which depicts both the nonlinear interaction and eventual decomposition into two separate single soliton solutions. 5 Parallel Surfaces. 101) is said to be parallel to .
Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory by C. Rogers;W. K. Schief