By C. Rogers;W. K. Schief

ISBN-10: 052181331X

ISBN-13: 9780521813310

This booklet describes the outstanding 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 by means of such eminent geometers as Bianchi, Darboux, Bäcklund, and Eisenhart on adjustments of privileged sessions of surfaces which depart key geometric homes unchanged. favorite among those are Bäcklund-Darboux variations with their impressive linked nonlinear superposition ideas and significance in soliton idea.

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**Example text**

Und des Schwerpunktes der Wellengruppe . . auf” [345, p 189]. 2 The Sine-Gordon Equation 23 Let be a pseudospherical surface with total curvature K = −1/ 2 and with generic position vector r = r(u, v), where u, v correspond to the parametrisation by arc length along asymptotic lines. In this parametrisation, ru , rv and N are all unit vectors, but ru and rv are not orthogonal. Accordingly, it proves convenient to introduce an orthonormal triad {A, B, C}, where (ru × rv ) , C=N sin = cosec rv − cot ru .

In the particular case when K = −1/ 2 < 0 is a constant, is termed a pseudospherical surface. 22) then yield a = a(u), b = b(v). 23) reduces to the celebrated sine-Gordon equation uv = 1 sin . 26) while those of Weingarten give 1 1 cot ru − cosec rv , 1 1 Nv = − cosec ru + cot rv . 27) In the twentieth century, the sine-Gordon equation has been shown, remarkably, to arise in a diversity of areas of physical interest (see [311]). It was the work of Seeger et al. [201, 345, 346] that first demonstrated how the classical B¨acklund transformation for this equation has important application in the theory of crystal dislocations.

52) descriptive of the action of the B¨acklund transformation B at the pseudospherical surface level. 3 Bianchi’s Permutability Theorem. 47) to construct multi-soliton solutions of the sine-Gordon equation. 25). 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.

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

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