§21.7(i) Connection of Riemann Theta Functions to RiemannSurfaces
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►Removing the singularities of this curve gives rise to a two-dimensional connected manifold with a complex-analytic structure, that is, a Riemannsurface. All compact Riemannsurfaces can be obtained this
way.
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►►►Figure 21.7.1: A basis of cycles for a genus 2 surface.
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§21.10(ii) Riemann Theta Functions Associated with a RiemannSurface
►In addition to evaluating the Fourier series, the main problem here is to compute a Riemann matrix originating from a Riemannsurface.
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Belokolos et al. (1994, Chapter 5) and references therein. Here the
Riemannsurface is represented by the action of a Schottky group on a region of
the complex plane. The same representation is used in Gianni et al. (1998).
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►He has worked on integrable systems, algorithms for computations with Riemannsurfaces, Bose-Einstein condensates, and methods to investigate the stability of solutions of nonlinear wave equations.
He is the coauthor of several Maple commands to work with Riemannsurfaces and the command to compute multidimensional theta functions numerically.
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►Particularly important for the use of Riemann theta functions is the Kadomtsev–Petviashvili (KP) equation, which describes the propagation of two-dimensional, long-wave length surface waves in shallow water (Ablowitz and Segur (1981, Chapter 4)):
…These parameters, including , are not free: they are determined by a compact, connected Riemannsurface (Krichever (1976)), or alternatively by an appropriate initial condition (Deconinck and Segur (1998)).
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►►►Figure 21.9.2: Contour plot of a two-phase solution of Equation (21.9.3).
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Magnify►Furthermore, the solutions of the KP equation solve the Schottky
problem: this is the question concerning conditions that a Riemann matrix needs to satisfy in order to be associated with a Riemannsurface (Schottky (1903)).
Following the work of Krichever (1976), Novikov conjectured that the Riemann theta function in (21.9.4) gives rise to a solution of the KP equation (21.9.3) if, and only if, the theta function originates from a Riemannsurface; see Dubrovin (1981, §IV.4).
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intersection index of and , two cycles lying on a closed
surface. if and do not intersect.
Otherwise gets an additive contribution from every
intersection point. This contribution is if the basis of
the tangent vectors of the and cycles
(§21.7(i)) at the point of intersection is
positively oriented; otherwise it is .
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►Figure 21.4.1 provides surfaces of the scaled Riemann theta function , with
…This Riemann matrix originates from the Riemannsurface represented by the algebraic curve ; compare §21.7(i).
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Figure 21.4.1:
parametrized by (21.4.1).
The surface plots are of , , (suffix 1); , , (suffix 2); , , (suffix 3).
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►Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: , , .
This Riemann matrix originates from the genus 3 Riemannsurface represented by the algebraic curve ; compare §21.7(i).
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C. L. Tretkoff and M. D. Tretkoff (1984)Combinatorial Group Theory, RiemannSurfaces and Differential Equations.
In Contributions to Group Theory,
Contemp. Math., Vol. 33, pp. 467–519.
JTEM (website)
Java Tools for Experimental Mathematics
ⓘ
Notes:
Single and double precision. Contains packages to compute elliptic functions, Riemann theta functions and their derivatives, and Riemann matrices for Riemannsurfaces.