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Computational Approach to
Riemann SurfacesC. Klein (Dijon)
withJ. Frauendiener (Dunedin)
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Theta-functional solutions to the Kadomtsev-Petviashvili equation
•3u
yy
+ �x
(6uux
+ uxxx
� 4ut
) = 0
weakly two-dimensional waves in shallow water
• almost periodic solutions in terms of theta functionson arbitrary compact Riemann surfaces
u = 2�2x
ln �(Ux + Vy + Wt + D) + 2c
(Krichever 1978)
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Hyperelliptic solutions (g=2)
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Hyperelliptic solutions (g=2)
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Hyperelliptic solution (g=4)
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Hyperelliptic solution (g=4)
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(Line)Solitons (localized in one direction), 2-soliton
✦ branch points coincide pairwise, surface of genus 0 in the limit
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(Line)Solitons (localized in one direction), 2-soliton
✦ branch points coincide pairwise, surface of genus 0 in the limit
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(Line)Solitons, 4-soliton
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(Line)Solitons, 4-soliton
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Symbolic vs. numerical
✦ Deconinck, v. Hoeij, Patterson: algcurves package in Maple (2001)
✦ symbolic approach, exact expressions (e.g. RootOf(x^2-2)) manipulated and numerically evaluated, in principle infinite precision
✦ Frauendiener, K.: fully numeric approach (floating point), hyperelliptic curves (1998), much more rapid, allows study of families of curves and of more complicated curves
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Outline✦ Riemann surfaces and algebraic curves✦ Branch points and singular points✦ Monodromy and homology✦ Puiseux expansions and holomorphic
differentials✦ Real Riemann surfaces✦ Hyperelliptic surfaces
✦ Performance tests and examplesDonnerstag, 19. September 2013
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Riemann surfaces
✦ Definition: A Riemann surface is a connected one-dimensional complex analytic manifold, i.e., a connected two-dimensional real manifold R with a complex structure Σ on it
✦ Theorem: All compact Riemann surfaces can be described as compactifications of non-singular algebraic curves
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Algebraic curves• Definition: plane algebraic curve C subset in C2
,
C = {(x, y) � C2|f(x, y) = 0),
f(x, y) =
MX
i=0
NX
j=0
aijxiy
j=
NX
j=0
aj(x)y
j
1
1 2 3 4 5 763
2
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Critical points
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1.5
−1
−0.5
0
0.5
1
1.5
x
y
simple double point:
y
2+ x
3 � x
2= 0
• general position: N distinct solutions y
n
for given x, N sheets of the
Riemann surface
• Implicit function theorem: unique solution to f(x, y) = 0 in vicinity of
solution (x0, y0) if f
y
(x0, y0) ⇥= 0
• branch point: f(x0, y0) = f
y
(x0, y0) = 0, but f
x
(x0, y0) ⇥= 0
singular point: f(x0, y0) = f
y
(x0, y0) = f
x
(x0, y0) = 0
• critical points given by the resultant R(x) of Nf � f
y
y and f
y
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Resultant
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Numerical root finding• construct companion matrix (has R(x) as the characteristic polynomial),
find eigenvalues with machine precision
• multiple zeros are not found with machine precision, ex. y
7= x(x � 1)
2
Klein curve, R(x) = x
6(x�1)
12, roots(R(x)) returns the following cluster
of roots
1.1053 + 0.0297i
1.1053 - 0.0297i
1.0736 + 0.0790i
1.0736 - 0.0790i
1.0224 + 0.1032i
1.0224 - 0.1032i
0.9686 + 0.0980i
0.9686 - 0.0980i
0.9264 + 0.0686i
0.9264 - 0.0686i
0.9037 + 0.0245i
0.9037 - 0.0245i,
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polynomial root finding✦ badly conditioned problem✦ Zeng: multroot package for multiple roots
(Newton iteration, minimize error by choice of multiplicity structure)
✦ resultant high order polynomial, therefore direct Newton iteration in x and y. Initial iterates from resultant with respect to x and y, pairing
✦ endgame for higher order zerosDonnerstag, 19. September 2013
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Singularities• multiple roots are tested for vanishing f
x
(x, y)
• infinity: homogeneous coordinates X,Y, Z
via x = X/Z, y = Y/Z
F (X,Y, Z) = Z
d
f(X/Z, Y/Z) = 0
infinite points: Z = 0, finite points: Z = 1
• Singular points at infinity:
F
X
(X,Y, 0) = F
Y
(X,Y, 0) = F
Z
(X,Y, 0) = 0
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Example• curve
f(x, y) = y
3+ 2x
3y � x
7= 0 ,
• finite branch points
bpoints =
-0.3197 - 0.9839i
0.8370 - 0.6081i
-1.0346
0
0.8370 + 0.6081i
-0.3197 + 0.9839i
• singularities,
sing =
0 0 1 4
0 1 0 9
corresponding to x = y = 0 and Y = 1, X = Z = 0
X Y Z
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Fundamental group
1
2
n-1
n
Υ1Υ2
Υn-1
Υn
Υ∞
• branching structure at critical points,
lift closed contours in the base around
points b1, . . . , bn to the covering
• generators {�k}nk=1 of
fundamental group
⇥1(CP1\{b1, . . . , bn})�1�2 . . . �n�1 = id
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Minimal spanning tree✦ Maple: halfcircles around critical points, deformation
of connecting paths
✦ shortest integration paths: start with critical point close to the base, choose point with minimal distance, iterate (Frauendiener, K, Shramchenko 2011)
✦ ex:
−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
4
1
2
3
5
6
−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
5
12
34
6
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✦ analytic continuation along a generator: sheets can change
✦ monodromy at infinity follows from condition on generators
Monodromies
• ex.:
Mon =
2 1 1 1 2 2 2
1 3 3 3 1 1 3
3 2 2 2 3 3 1
1
1 2 3 4 5 763
2
−1−0.5
00.5
1
−1
−0.5
0
0.5
1−1
−0.5
0
0.5
1
−1−0.5
00.5
1
−1
−0.5
0
0.5
1−1
−0.5
0
0.5
1
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Homology
✦ Tretkoff-Tretkoff algorithm: Riemann surface connected, planar tree for given monodromies
✦ 2g+N-1 closed contours built from the generators of the fundamental group, with known intersection numbers
✦ canonical basis of the homology: ai � bj = �bj � ai = �ij i, j = 1, . . . , g
Q5
A2
B6
B5
B2
B4
A2
B7
B1
B3
A2
A1
B6
B7
B3B1
B7
B5
A2
A3
A2
A3
P1
P2
P3
P4
P5
P6
Q4
Q2
Q3Q1
A3
Q6
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Puiseux expansion
• desingularization: atlas of local coordinates to identify
all sheets in the vicinity of the singularity
• y2= x, no Taylor expansion y(x) near (0, 0), Puiseux expansion
x = tr , y = �1ts1
+ ...
r, s1, . . . � N, �i � C for i = 1, 2, . . .
• y = 0 zero of order m for f(0, y) = 0, m inequivalent expansions
needed to identify all sheets, singular part
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Newton polygon
• ex.: f(x, y) = y
3+ 2x
3y � x
7= 0
PuiExp{1} =
2.0000 3.0000 0 + 1.4142i
2.0000 3.0000 0 - 1.4142i
1.0000 4.0000 0.5000
PuiExp{2} =
4.0000 7.0000 -1.0000
4.0000 7.0000 0 + 1.0000i
4.0000 7.0000 0 - 1.0000i
4.0000 7.0000 1.0000
PuiExp{1} for (0, 0) ([0, 0, 1]), PuiExp{2} for infinity ([0, 1, 0])
3
1
3
7
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Holomorphic 1-forms• holomorphic in each coordinate chart, g-dimensional space
• Noether:
⇥k =
Pk(x, y)
fy(x, y)
dx ,
adjoint polynomials Pk(x, y) =
Pi+jd�3 c(k)
ij xiyj,
degree at most d� 3 in x and y (d = max(i + j) for aij ⇥= 0)
• singular point P : �P conditions via Puiseux expansions
• infinity: homogeneous coordinates
• ex.: f(x, y) = y3+ 2x3y � x7
= 0
⇥1 =
x3
3y2+ 2x3
, ⇥2 =
xy
3y2+ 2x3
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Cauchy integral approach• numerical problem: cancellation errors, ex.
e
x�1x
for x! 0
• Cauchy formula
f(t) =
1
2⇡i
Z
�
f(t
0)
t
0 � t
dt
0
• closed contours around critical points identified via monodromy group
• series in t for holomorphic f (|t| < |t0|)
f(t) =
1
2⇡i
1X
n=0
t
n
Z
�
f(t
0)
dt
0
(t
0)
n+1
• Puiseux series: f = y;
holomorphicity condition for di↵erentials (no negative powers)
• infinity: express �1 in terms of the �
i
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Numerical integration• Gauss-Legendre integration: expansion of integrand in terms of
Legendre polynomials F(xl) =PNl
k=0 akPk(xl), l = 0, . . . , Nl
Z 1
�1F(x)dx �
NlX
k=0
ak
Z 1
�1Pk(x)dx
• integration:Z 1
�1F(x)dx �
NlX
k=0
F(xk)Lk
• analytic continuation of yj along the �i on the collocationpoints xl, integration of the holomorphic di�erentials
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Riemann matrix
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Performance
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
−14
−12
−10
−8
−6
−4
log2 Nl
log 10
err
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−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
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Theta functions
• theta series approximated as sum
�(z|B) ⇥N�X
N1=�N�
. . .N�X
Ng=�N�
expn
i⇤D
B ⌅N, ⌅NE
+ 2⇤iD
⌅z, ⌅NEo
• use periodicity properties of theta function to have argument in the funda-mental cell, ⇥1 smallest eigenvalue of the imaginary part of the Riemannmatrix
N� >12
+r
14� ln �
⇤⇥1
• symplectic transformation to Siegel fundamental domain (approximately)(Deconinck et al. 2002)
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−1.5 −1 −0.5 0 0.5 1 1.5−1.5
−1
−0.5
0
0.5
1
1.5
4
1
2
3
5
6
✦ A(P): determine closest marked point to P, analytic continuation of y from there and integration as before.
✦ critical points, infinity: substitution as indicated by Puiseux expansions
Abel map
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Real Riemann surfaces✦ in applications, solutions to PDEs in terms of
theta functions must satisfy reality and smoothness conditions
✦ real Riemann surfaces: anti-holomorphic involution, convenient form of the homology basis
✦ smoothness: study of the theta divisor (zeros of the theta function) (Dubrovin, Natanzon, Vinnikov)
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Davey-Stewartson equations
✦ model the evolution of weakly nonlinear water waves traveling predominantly in one direction, wave amplitude slowly modulated in two horizontal directions, plasma physics, ...
✦ completely integrable, theta-functional solutions (Malanyuk 1994, Kalla 2011)
✦ algorithm to transform computed homology basis to `Vinnikov‘ basis (K, Kalla 2011)
i⇤t
+ ⇤xx
� �2 ⇤yy
+ 2 (� + ⇥ |⇤|2) ⇤ = 0,
� = i, 1, ⇥ = ±1, �xx
+ �2 �yy
+ 2⇥ |⇤|2xx
= 0,
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Trott curve
DS1+, �(a) = �0.2, �(b) = 0.2t 2 [�2, 2]
• M-curve, g = 3, real simple branch points (s = (1,�1,�1))
144 (x
4+ y
4)� 225 (x
2+ y
2) + 350 x
2y
2+ 81 = 0
� = i, ⇥ = 1
Donnerstag, 19. September 2013
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Trott curve
DS1+, �(a) = �0.2, �(b) = 0.2t 2 [�2, 2]
• M-curve, g = 3, real simple branch points (s = (1,�1,�1))
144 (x
4+ y
4)� 225 (x
2+ y
2) + 350 x
2y
2+ 81 = 0
� = i, ⇥ = 1
Donnerstag, 19. September 2013
![Page 40: Computational Approach to Riemann Surfacesfrompde.sissa.it/workshop2013/talks/18Wed/Klein.pdf · Riemann surfaces and algebraic curves Branch points and singular points Monodromy](https://reader035.vdocuments.us/reader035/viewer/2022070909/5f93d37ef7917f7c010cc3c6/html5/thumbnails/40.jpg)
DS on Fermat curve g=3DS2�, �(a) = �1.5 + i, �(a) = �1.5� i t 2 [�5, 5]
Donnerstag, 19. September 2013
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DS on Fermat curve g=3DS2�, �(a) = �1.5 + i, �(a) = �1.5� i t 2 [�5, 5]
Donnerstag, 19. September 2013
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Hyperelliptic surfaces ✦ general surface: analytic continuation most time consuming
✦ y: square root of polynomial in x, holomorphic differentials known,
✦ analytic continuation of the root trivial (square root, correct unwanted sign changes)
✦ branch points prescribed, can almost collapse
✦ homology can be chosen a priori
1F 2E 2F g+1E g+1F1E
1b1a ga
gb
y
2 +Q2g+2
i=1 (x� xi) = 0
Donnerstag, 19. September 2013
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Outlook
✦ more efficient determination of critical points, homotopy tracing, endgame
✦ Abel map via Cauchy formula✦ Siegel transformation of the Riemann matrix
to fundamental domain✦ parallelization of theta functions
Donnerstag, 19. September 2013
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References• C. Kalla and C. Klein, Computation of the topological type of a real Rie-
mann surface (2012) arXiv:1204.4826
• C. Kalla and C. Klein, On the numerical evaluation of algebro-geometricsolutions to integrable equations, Nonlinearity Vol. 25 569-596 (2012).
• J. Frauendiener, C. Klein and V. Shramchenko, E�cient computation ofthe branching structure of an algebraic curve, Comput. Methods Funct.Theory Vol. 11 (2011), No. 2, 527–546.
• J. Frauendiener and C. Klein, Algebraic curves and Riemann surfaces inMatlab, in A. Bobenko and C. Klein (ed.), Computational Approach toRiemann Surfaces, Lecture Notes in Mathematics Vol. 2013 (Springer)(2011).
• J. Frauendiener and C. Klein, Hyperelliptic theta functions and spectralmethods: KdV and KP solutions, Lett. Math. Phys., Vol. 76, 249-267(2006).
• J. Frauendiener and C. Klein, Hyperelliptic theta functions and spectralmethods, J. Comp. Appl. Math., Vol. 167, 193 (2004).
• C. Klein and O. Richter, Ernst Equation and Riemann Surfaces, LectureNotes in Physics Vol. 685 (Springer) (2005).
Donnerstag, 19. September 2013