Download Algebraic geometry 03 Complex algebraic varieties, Algebraic by A.N. Parshin (editor), I.R. Shafarevich (editor), I. Rivin, PDF

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.

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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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