By A.N. Parshin (editor), I.R. Shafarevich (editor), I. Rivin, V.S. Kulikov, P.F. Kurchanov, V.V. Shokurov

ISBN-10: 3540546812

ISBN-13: 9783540546818

This two-part EMS quantity presents a succinct precis of advanced algebraic geometry, coupled with a lucid creation to the new paintings at the interactions among the classical region of the geometry of complicated algebraic curves and their Jacobian types. a very good spouse to the older classics at the topic.

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**Extra resources for Algebraic geometry 03 Complex algebraic varieties, Algebraic curves and their Jacobians**

**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.

### Algebraic geometry 03 Complex algebraic varieties, Algebraic curves and their Jacobians by A.N. Parshin (editor), I.R. Shafarevich (editor), I. Rivin, V.S. Kulikov, P.F. Kurchanov, V.V. Shokurov

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