C. Rogers;W. K. Schief's Bäcklund and Darboux Transformations: Geometry and Modern PDF

By C. Rogers;W. K. Schief

ISBN-10: 052181331X

ISBN-13: 9780521813310

This booklet describes the amazing connections that exist among the classical differential geometry of surfaces and smooth soliton idea. The authors additionally discover the large physique of literature from the 19th and early 20th centuries by means of such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on variations of privileged periods of surfaces which depart key geometric homes unchanged. in demand among those are Bäcklund-Darboux changes with their striking linked nonlinear superposition rules and value in soliton thought.

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Additional resources for Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory

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N-soliton solutions of the sine-Gordon equation may be thereby generated by purely algebraic procedures. 54) with B¨acklund parameters ␤ = ␤1 , . . , ␤N . Thus, at each application of the B¨acklund transformation, a new B¨acklund parameter ␤i is introduced and an ith order soliton generated. 3. 2 Physical Applications Seeger et al. [345] exploited the permutability theorem for the sine-Gordon equation to investigate interaction properties of kink and breather-type solutions in connection with a crystal dislocation model.

4 Pseudospherical Soliton Surfaces. 88) where ␤ = tan(␨ /2). 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.

25). Let ␻ denote the B¨acklund transform of ␻ via B␤ . 52) where L = ␳ sin ␨. 26). 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.

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Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory by C. Rogers;W. K. Schief

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