By C. Rogers, W. K. Schief
This publication describes the striking connections that exist among the classical differential geometry of surfaces and glossy soliton concept. The authors additionally discover the huge physique of literature from the 19th and early 20th centuries via such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on differences of privileged sessions of surfaces which depart key geometric houses unchanged. popular among those are Bäcklund-Darboux ameliorations with their notable linked nonlinear superposition rules and significance in soliton idea.
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Extra info for Backlund & Darboux Transformations
25) and leaves it invariant. 25) holds: otherwise the compatibility condition uv = vu is not satisﬁed. • B␤ contains a parameter ␤ = tan( /2) injected into the underlying Bianchi transformation by a Lie group invariance. 26) associated with pseudospherical surfaces parametrised by arc length along asymptotic lines. 26) induces the B¨acklund transformation B␤ operating at the nonlinear level. In that B␤ represents a correspondence between solutions of the same equation, it is commonly termed an auto-B¨acklund transformation.
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. 91), where i = 1 2␤i 1 + ␤i2 x − 1 − ␤i2 y . 4 Pseudospherical Soliton Surfaces. 96) with = [1 − (c2 + d 2 )]x − [1 + (c2 + d 2 )]y and = [1 + (c2 + d 2 )]x − [1 − (c2 + d 2 )]y.
If we write d = p/q, where p and q are co-prime integers with p < q, then the period of the breather solution is 2q/ p. 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.
Backlund & Darboux Transformations by C. Rogers, W. K. Schief