Integrability in Discrete Differential Geometry: From DDG ... · Discretization Principles I...
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Integrability in Discrete Differential Geometry:From DDG to the classification of discrete
integrable systems
Alexander Bobenko
Technische Universität Berlin
LMS Summer School, Durham, July 2016
CRC 109 “Discretization in Geometry and Dynamics”
Alexander Bobenko DDG and Classification of discrete integrable equations
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Discrete Differential Geometry
I Aim: Development of discrete equivalents of the geometricnotions and methods of differential geometry. The latterappears then as a limit of refinements of the discretization.
I Question: Which discretization is the best one?I (Theory): preserves fundamental properties of the smooth
theoryI (Applications): represent smooth shape by a discrete shape
with just few elements; best approximation
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Surfaces and transformations
Classical theory of (specialclasses of) surfaces (constantcurvature, isothermic, etc.)
General and specialQuad-surfaces
special transformations(Bianchi, Bäcklund, Darboux)
discrete→ symmetric
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Basic idea
Do not distinguish discrete surfaces and their transformations.Discrete master theory.
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Example - planar quadrilaterals as discrete conjugate systems.Multidimensional Q-nets [Doliwa-Santini ’97].
Alexander Bobenko DDG and Classification of discrete integrable equations
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Basic idea
Do not distinguish discrete surfaces and their transformations.Discrete master theory.
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Example - planar quadrilaterals as discrete conjugate systems.Multidimensional Q-nets [Doliwa-Santini ’97].
Alexander Bobenko DDG and Classification of discrete integrable equations
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Discretization Principles
I Transformation Group Principle. Smooth geometricobjects and their discretizations belong to the samegeometry, i.e. are invariant with respect to the sametransformation group(discrete Klein’s Erlangen Program)
I Consistency Principle. Discretizations of smoothparametrized geometries can be extended tomultidimensional consistent nets(Integrability)
Multidimensional Q-nets (projective geometry) can be restrictedto an arbitrary quadric (⇒ Discretization of classicalgeometries) [Doliwa ’99]
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Differential geometry
Smooth limit:I Differential geometry follows from incidence theorems of
projective geometry
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Integrability as Consistency
I Equation I Consistency
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a b
c d
Q(a,b, c,d) = 0
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Integrability as Consistency
I Equation I Consistency
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a b
c d
Q(a,b, c,d) = 0
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Integrability as Consistency
I Equation I Consistency
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a b
c d
Q(a,b, c,d) = 0
Alexander Bobenko DDG and Classification of discrete integrable equations
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Integrability as Consistency
I Equation I Consistency
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a b
c d
Q(a,b, c,d) = 0
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Why integrability?
Can be derived from consistency:I Lax representationI Bäcklund-Darboux transformations
Bobenko-Suris [’02], Nijhoff [’02]
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Book in DDG
A.I. Bobenko, Y.B. Suris,Discrete differential geometry.Integrable structure.AMS, Graduate Studies inMathematics, v.98, 2008,xxiv+404 pp.
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Classification 2D
I hyperbolic nonlinear equation Q(a,b, c,d) = 0I Q multi-affine (can be resolved with respect to any
variable)I Classification of integrable (i.e. consistent) equations.
[Adler, B., Suris ’03]
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Classification. Method’s overview
-
6
m
n
Qm,n(xm,n, xm+1,n, xm,n+1, xm+1,n+1) = 0
Alexander Bobenko DDG and Classification of discrete integrable equations
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Classification. Method’s overview
-
6
m
n
Qm,n(xm,n, xm+1,n, xm,n+1, xm+1,n+1) = 0
-integrability =
3D-consistency ��
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Alexander Bobenko DDG and Classification of discrete integrable equations
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Classification. Method’s overview
-
6
m
n
Qm,n(xm,n, xm+1,n, xm,n+1, xm+1,n+1) = 0
-integrability =
3D-consistency ��
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?
analysis ofsingular solutions
list of integrable equations
Alexander Bobenko DDG and Classification of discrete integrable equations
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Classification. Method’s overview
-
6
m
n
Qm,n(xm,n, xm+1,n, xm,n+1, xm+1,n+1) = 0
-integrability =
3D-consistency ��
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?
analysis ofsingular solutions
list of integrable equations
assumptions: PPPPPPPPPqmulti-affine Q
Alexander Bobenko DDG and Classification of discrete integrable equations
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Classification. Method’s overview
-
6
m
n
Qm,n(xm,n, xm+1,n, xm,n+1, xm+1,n+1) = 0
-integrability =
3D-consistency ��
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?
analysis ofsingular solutions
list of integrable equations
assumptions: PPPPPPPPPqmulti-affine Q
+ some nondegeneracy condition
Alexander Bobenko DDG and Classification of discrete integrable equations
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3D-consistency
0 1
2 12
3 13
23 123 3D-consistency: the values ofx123 computed in 3 possibleways coincide identically on theinitial values x , x1, x2, x3.
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3D-consistency
0 1
2 12
3 13
23 123 3D-consistency: the values ofx123 computed in 3 possibleways coincide identically on theinitial values x , x1, x2, x3.
Alexander Bobenko DDG and Classification of discrete integrable equations
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3D-consistency
0 1
2 12
3 13
23 123 3D-consistency: the values ofx123 computed in 3 possibleways coincide identically on theinitial values x , x1, x2, x3.
Alexander Bobenko DDG and Classification of discrete integrable equations
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Singular solutions
We consider only multi-affine equations (= of the first degree oneach unknown):
Q(x , y , z, t) = a1xyzt + · · ·+ a16 = 0. (1)
The important role play biquadratic curves:
h(x , y) = QzQt −QQzt = h1x2y2 + · · ·+ h9 = 0.
We associate such curve to each edge of thesquare cell Q
6
?
� -
x y
t z
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Biquadratics
The key observation is given by the following theorem.
Theorem. Let the equations be 3D-consistent and all involved biquadraticsbe not degenerate. Then, for each edgeof the cube, the equations correspondingto adjacent faces give rise to one and thesame biquadratic curve.
The nondegeneracy assumption means that a biquadraticpolynomial h(x , y) must be free of the factors of the formx − const and y − const =⇒ two types of equations.
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Singular solutions
The idea of the proof.
Choose the singular initial data on the face (1,2). This leads toan undetermined value of x123. However, due to consistency,x123 can be found without using this face. Therefore, the initialdata on the faces (1,3) and (2,3) must be singular as well.Therefore, the singular curves on these faces have the sameprojections on the common edges.
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Möbius transformations
The classification is made modulo (PSL2(C))8, that is thevariables in all vertices of the cube are subjected toindependent Möbius transformations.
It is important that the following commutative diagram iscompatible with the action of this group.
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Möbius transformations
r4(x4)δ3←−− h34(x3, x4)
δ4−−→ r3(x3)
δ1
x xδ12
xδ2
h14(x1, x4)δ23←−− Q(x1, x2, x3, x4)
δ14−−→ h23(x2, x3)
δ4
y yδ34
yδ3
r1(x1)δ2←−− h12(x1, x2)
δ1−−→ r2(x2)
where
δij(Q) = Qxi Qxj −QQxi ,xj , δi(h) = h2xi− 2hhxi ,xi .
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List of 2D integrable equations
Theorem. Up to Möbius transformations, any 3D-consistentsystem with nondegenerate biquadratics is one of the followinglist (α = α(i), β = α(j), sn(α) = sn(α; k)):
α(x − xj)(xi − xij)− β(x − xi)(xj − xij) = δαβ(β − α) (Q1)
α(x − xj)(xi − xij)− β(x − xi)(xj − xij)
+αβ(α− β)(x + xi + xj + xij)
= αβ(α− β)(α2 − αβ + β2)
(Q2)
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List of 2D integrable equations
Theorem. Up to Möbius transformations, any 3D-consistentsystem with nondegenerate biquadratics is one of the followinglist (α = α(i), β = α(j), sn(α) = sn(α; k)):
(α− 1
β
)(xxi + xjxij)−
(β − 1
α
)(xxj + xixij)
−(αβ− β
α
)(xxij + xixj)
− δ4
(α− 1
α
)(β − 1
β
)(αβ− β
α
)= 0,
(Q3)
sn(α) sn(β) sn(α− β)(k2xxixjxij + 1) + sn(α)(xxi + xjxij)
− sn(β)(xxj + xixij)− sn(α− β)(xxij + xixj) = 0.(Q4)
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List of 2D integrable equations
Q1 [Quispel-Nijhoff-Capel-Van der Linden ’84]Q2 [Adler-Bobenko-Suris ’03]
Q3δ=0 [Quispel-Nijhoff-Capel-Van der Linden ’84]Q3δ 6=0 [Adler-Bobenko-Suris ’03]
Q4 [Adler ’98]
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List of 2D integrable equations
Equations with degenerated biquadratics:
(x − xij)(xi − xj) = α− β (H1)
(x − xij)(xi − xj) + (β − α)(x + xi + xj + xij) = α2 − β2 (H2)
α(xxi + xjxij)− β(xxj + xixij) = δ(β2 − α2) (H3)
Alexander Bobenko DDG and Classification of discrete integrable equations