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Triangulations and soliton graphs
Rachel Karpman and Yuji Kodama
The Ohio State University
September 4, 2018
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Introduction
Outline
Setting: The KP equation.
Soliton graphs.
Duality and soliton triangulations.
Results.
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The KP equation
The KP equation
Non-linear dispersive wave equation
∂
∂x
(−4
∂u
∂t+ 6u
∂u
∂x+∂3u
∂x3
)+ 3
∂2u
∂y2= 0.
Line-soliton solutions of the KP equation model shallow-water waveswith peaks localized along straight lines.
Combinatorics of KP solitons studied in (?).
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The KP equation
The Grassmannian
Regular line-soliton solutions can be constructed from points in thetotally nonnegative Grassmannian.
For N ≤ M, let Gr(N,M) be the Grassmannian of N-planes inM-space.
Represent points in Gr(N,M) by full-rank N ×M matrices, modulorow operations.
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The KP equation
Plucker coordinates
The N × N matrix minors give homogeneous coordinates onGr(N,M), called Plucker coordinates.
For I an N-element subset of {1, 2, . . . ,M}, let ∆I (A) denote thematrix minor of A corresponding to columns indexed by I .
The collection N-tuples indexing non-vanishing Plucker coordinates ofA the matroid of A, denoted M(A).
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The KP equation
The totally nonnegative Grassmannian
The totally nonnegative Grassmannian Gr≥0(N,M) is the locus ofGr(N,M) where all Plucker coordinates are nonnegative.
Similarly, the totally positive Grassmannian Gr>0(N,M) is the locuswhere all Plucker coordinates are positive.
The combinatorics of Gr≥0(M,N) was first studied in (?).
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The KP equation
Notation
Fix M “generic enough” points (pi , qi ) on the parabola q = p2, with
p1 < p2 < · · · < pM .
For I = {i1 < · · · < iN} ∈([M]
N
), let
ΘI =∑i∈I
pix + qiy + ωi (t).
Here t is the multi-time parameter (t3, . . . , tM) and
ωi (t) =M−1∑k=3
pki tk .
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The KP equation
From matrices to soliton solutions
Let KI =∏
`<m(pim − pi`)
Let A be a matrix representing a point in Gr≥0(N,M).
A uA(x , y , t) = 2∂2
∂x2(ln τA(x , y , t))
where we haveτA =
∑I∈M(A)
∆I (A)KI exp(ΘI )
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Soliton Graphs
Contour plots
The function uA(x , y , t) models the height of a wave at time t.
Wave peaks give a contour plot.
100
200
-100
-200
0
100
200
-100
-200
0
-200 -100 100 2000 -200 -100 100 2000
t = 70t = 0
100
200
-100
-200
0
-200 -100 100 2000
t = -70
Figure: Contour plots corresponding to a point in Gr≥0(3, 6)
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Soliton Graphs
Contour plots are tropical curves
Can approximate the contour plot as the locus where
fA = max {ln(∆I (A)KI ) + ΘI : I ∈M(A)}
is nonlinear.
So fA chops up the (x , y)-plane into regions where one plane
z = ln (∆I (A)KI ) + ΘI (x , y , t)
is dominant over the others.
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Soliton Graphs
Asymptotic contour plots
Asymptotic contour plots: rescale the variables, can assume thescalars ∆I (A) are negligible.
The asymptotic contour plot for fixed multi-time parameter t0 is thelocus where
fM = max{ΘI (x , y , t0) | I ∈M(A)}
is non-linear.
More precisely, fM = lims→∞1s fA(sx , sy , st0).
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Soliton Graphs
Example
Let N = 1, M = 4.
Choose parameters
p1 = −2 p2 = 0 p3 = 1 p4 = 2.
Let t = t3 = 1.
Then we have
θ1(x , y , t) = −2x + 4y − 8
θ2(x , y , t) = 0
θ3(x , y , t) = x + y + 1
θ4(x , y , t) = 2x + 4y + 8
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Soliton Graphs
Example Continued
-7 -6 -5 -4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
0
θ1
θ2
θ3
θ4
x
y
Figure: An asymptotic contour plot.
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Soliton Graphs
Soliton graphs
Goal: understand combinatorics of asymptotic contour plots.
Take plots up to isotopy, get soliton graphs.
Restrict to Gr>0(N,M), where all Plucker coordinates positive.
In this case, soliton graphs are plabic graphs (?).
Want to classify soliton graphs for Gr>0(N,M).
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Soliton Graphs
Constructing soliton graphs
Embed asymptotic contour plot in a disk, take graph up to isotopy.
Color an internal vertex white if the adjoining regions share M − 1indices, and black otherwise.
2 4
1 4
1 2 3 4
2 3
2 4
1 4
1 2 3 4
2 3
Figure: From contour plot to soliton graph.
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Soliton Graphs
Plabic graphs
Planar, bicolored graph embedded in a disk, satisfies some technicalconditions.
We label each face F plabic graph with an i if F is to the left of thezig-zag path Ti ending at boundary vertex i .
1
43
2
2 4
1 4
1 2 3 4
2 3
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Soliton Graphs
Face labels of plabic graphs
The face labels determine the graph, up to contracting andun-contracting unicolored edges.
Face labels of plabic graphs give clusters in the cluster algebrastructure of Gr(N,M) (?).
Every cluster containing only Plucker variables comes from a plabicgraph.
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Soliton Graphs
Weak Separation
A collection C of N-element subsets of {1, 2, . . . ,M} is the set of facelabels of a plabic graph for Gr(N,M) if and only if it is a maximalweakly separated collection.
For any I , J ∈ C, if the numbers 1, . . . ,M are arranged in order arounda circle, we can draw a chord that separates I\J from J\I .
C has N(M − N) + 1 elements (so it is as large as possible).
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Soliton Graphs
Realizability
Theorem (?)
Every soliton graph for Gr>0(N,M) is a plabic graph. Face labels of theplabic graph correspond to dominant exponentials of the soliton solution.
We say a collection of face labels is realizable if it comes from asoliton graph.
Goal: classify realizable collections.
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Duality and soliton triangulations
The duality map
Map a plane to a point:
θi (x , y) = pix + qiy + ωi 7→ vi = (pi , qi , ωi )
Take convex hull of points, project from above to (p, q)-plane.
Get triangulation of the M-gon.
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Duality and soliton triangulations
Example continued
Recall:
p1 = −2 p2 = 0 p3 = 1 p4 = 2 t = 1
-7 -6 -5 -4 -3 -2 -1 1 2 3 4
-4
-3
-2
-1
1
2
3
0
θ1
θ2
θ3
θ4
x
y
-2 -1 1 2
1
2
3
4v1
v2
v3
v4
x
y
0
Figure: A soliton tiling.
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Duality and soliton triangulations
The N = 2 case
ΘI (x , y) 7→ vI =∑i∈I
vi
-2 -1 1 2
1
2
3
4v1
v2
v3
v4
x
y
0
-2 -1 1 2 3 4
1
2
3
4
5
6
7
8
v2 4v1 2
v2 3
v3 4
v1 4
x
y
0
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Duality and soliton triangulations
Induction
Use induction algorithm to construct tiling for Gr>0(N + 1,M) fromtiling for Gr>0(N,M) (?).
1
2
3
4
5 6
7
8
9
10
11
2
1
5
2
9
11
8
6
4
1
2
3
4
5 6
7
8
9
10
11
Figure: Using the induction algorithm.
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Duality and soliton triangulations
Induction continued
Triangulation of the white polygons depends on the weights of ourpoints.
1 2
2 3
3 4
4 5
5 6
6 7
7 8
8 9
9 10
10 111 11
1 2 3
2 3 4
3 4 5
4 5 6 5 6 7
6 7 8
7 8 9
8 9 10
9 10 11
1 10 11
1 2 11
2 5 6
2 5 8
1 2 8
1 2 9
6 8 9
1 2 5
1 2 4
2 4 5
5 6 8
1 5 8
1 9 11
1 8 9
1 6 8
1 2 8
8 9 11
1 2 3
2 3 4
3 4 5
4 5 6 5 6 7
6 7 8
7 8 9
8 9 10
9 10 11
1 10 11
1 2 11
2 5 6
2 5 8
1 2 8
1 2 9
6 8 9
1 2 5
1 2 4
2 4 5
5 6 8
1 5 8
1 9 11
1 8 9
1 6 8
1 2 8
8 9 111 9 9 11
1 8
1 5
2 5
5 8
6 8
2 4
Figure: From N = 2 to N = 3.
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Results
Summary of results
Question: is every maximal weakly separated collection realizable?
Answer is yes for...
Gr>0(2,M) (?).
Gr>0(3, 6) (?).
Gr>0(3, 7),Gr>0(3, 8) [Kodama and K.]
In general, the answer is no.
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Results
Choices of parameters
For N = 3, M = 6, 7 or 8, every weakly separated collection isrealizable for some choice of parameters p1, p2, . . . , pM .
Which plabic graphs we can realize depends on our choice of pi .
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Results
Examples
1
2
34
5
61
2
3
4
5
6
136
235 145
135
123
234
345
456
156
126
134 356
125
135
123
234
345
456
156
126
Figure: Triangulations which are only realizable for some choices of parameters.
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Results
Classification for Gr(3, 6) and Gr(3, 7).
For Gr(3, 6), there are 34 possible graphs, each generic choice ofparameters lets us realize 32 of them (?).
For Gr(3, 7), there are 259 possible graphs, a each generic choice ofpi we can realize 231 of them [Kodama and K].
Main obstacle: same as in Gr(3, 6) case.
For Gr(3, 8), don’t yet have classification.
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Results
The general case
Not all weakly separated collections are realizable.
Can build plabic graph from any simple, non-stretchable arrangementof pseudolines, which gives a counter-example [Thomas, 2017].
Smallest counter-example of this form is for Gr(9, 18)
Conjecture: much smaller counter-examples exist.
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Results
References I
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