soliton decomposition of the box-ball systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf ·...
TRANSCRIPT
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Soliton decomposition of the Box-Ball System
Leonardo T. Rolla
with Pablo A. Ferrari, Chi Nguyen, Minmin Wang
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ResourcesSimulationhttps://mate.dm.uba.ar/˜leorolla/simulations/bbs.html
(on a 1d torus wrapped around a 2d torus)
These slideshttp://mate.dm.uba.ar/˜leorolla/bbs-slides.pdf
Articlehttps://arxiv.org/abs/1806.02798
(and references therein)
Extended abstracthttp://mate.dm.uba.ar/˜leorolla/bbs-abstract.pdf
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Ball-Box System (Takahashi-Satsuma 1990)
Ball configuration η ∈ {0, 1}Z
η(x) = 0↔ empty box, η(x) = 1↔ ball at x
Carrier picks balls from occupied boxes
Carrier deposits one ball, if carried, at empty boxes
0 0 1 0 1 1 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 η
0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 1 1 0 0 0 0 Tη
0 0 0 0 1 0 0 0 1 1 0 0 0 0 1 0 0 1 1 1 0 T 2η
Tη : configuration after the carrier visited all boxes.
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Formal definition
We say that x is an excursion point if, for some z 6 y,x∑y=z
η(y) >x∑y=z
[ 1− η(y) ],
otherwise x is a record.
Now we define
Tη(x) ={
0, x is a record,1− η(x), otherwise.
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Example
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Motivation: Korteweg & de Vries equation
u̇ = u′′′ + uu′
Soliton: a solitary wave that propagates with little loss of energy andretains its shape and speed after colliding with another such wave
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Take-home messages
Ergodic Theory ↔ Integrable System ↔ Algebraic Structures?
Identifying solitons and hierarchical structures
Interaction ; asymptotic speeds
Many conservations ; many invariant measures
Complete description of invariant measures still missing
Uniqueness of solutions to speed equations still missing
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Solitons in the BBS
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Outline of the talk
1) Conservation of k-solitons and how to identify them (T&S)
2) Asymptotic speed of k-solitons
3) k-slots and k-components
4) Invariant measures for T from independent k-components
5) Evolution of k-components is a hierarchical translation
6) Reconstruction from k-components
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Solitons
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Conservation of solitons
k-soliton: set of k successive ones followed by k zeros (for now)
Isolated k-solitons travel at speed k and conserve shape and distance:
000001110000000000000000001110000000000000000000001110000000000000000001110000000000000000000001110000000000000000001110000000000000000000001110000000000000000001110000000000000000000001110000000000000000001110000000000000000000001110000000000000000001000000000000000000000001110000000000000000000000000000000000000000001110000000000000000000000000000000000000000001110000000000000000000000000000000000000000001110000000000000000000000000000000000000000001110000000000000000000000000000000000000000001110
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Conservation of solitons
k-solitons and distances are conserved after interacting with m-solitons:
000001110000001000000000000111000000000000000000000000000000001110000100000000000000111000000000000000000000000000000001110010000000000000000111000000000000000000000000000000001101100000000000000000111000000000000000000000000000000010011100000000000000000111000000000000000000000000000001000011100000000000000000011100000000000000000000000000100000011100000000000000000011100000000000000000000000010000000011100000000000000000011100000000000000000000001000000000011100000000000000000011000000000000000000000100000000000011100000000000000000000000000000000000000010000000000000011100000000000000
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Conservation of solitons
Isolated k-solitons travel at speed k and conserve the distances:
.....111000...............111000..........
........111000...............111000.......
...........111000...............111000....
..............111000...............111000.
k-solitons and distances are conserved after interacting with m-solitons:
.....111000...10............111000...................
........111000.10..............111000................
...........11100100...............111000.............
..............11011000..............111000...........
................10.111000..............111000........
.................10...111000..............111000.....
..................10.....111000..............111000..
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Conservation of solitons
The following manual simulation shows the interaction between two k-poles configurations for k = 1 and k=2.
The red configuration is consists of monopoles with effective interpole distances (counted by dots, which are in fact records) 2, 3, 1, 5The blue configuration consists on bi-poles with effective distances 3, 4, 2, 1, 0, 3
The goal is to reconstruct those effective distances from any of the configurations in the middle. And better, to construct directly the configurations in the middle using only the effective distances.
Then, extend to any number of k-poles...
Observe that the interpole distances between k-poles are conserved. Need to show this.
....1100...1100....1100..1100.11001100...1100...1010..10...10.10.....10.........................................................................
......1100...1100....1100..1100.11001100...1100..1010..10...10.10.....10........................................................................
........1100...1100....1100..1100.11001100...1100.1010..10...10.10.....10.......................................................................
..........1100...1100....1100..1100.11001100...11001010..10...10.10.....10......................................................................
............1100...1100....1100..1100.11001100...11010100.10...10.10.....10.....................................................................
..............1100...1100....1100..1100.11001100...1010110010...10.10.....10....................................................................
................1100...1100....1100..1100.11001100..1010.110100..10.10.....10...................................................................
..................1100...1100....1100..1100.11001100.1010..101100.10.10.....10..................................................................
....................1100...1100....1100..1100.110011001010..10.110010.10.....10.................................................................
......................1100...1100....1100..1100.110011010100.10..11010010.....10................................................................
........................1100...1100....1100..1100.11001010110010...10110100....10...............................................................
..........................1100...1100....1100..1100.11010100110100..10.101100...10..............................................................
............................1100...1100....1100..1100.10101100101100.10.10.1100..10.............................................................
..............................1100...1100....1100..11001010.110100110010.10..1100.10............................................................
................................1100...1100....1100..11010100.10110011010010...110010...........................................................
..................................1100...1100....1100..1010110010.110010110100...110100.........................................................
....................................1100...1100....1100.1010.110100.110100101100...101100.......................................................
......................................1100...1100....11001010..101100.101101001100..10.1100.....................................................
........................................1100...1100....11010100.10.110010.1011001100.10..1100...................................................
..........................................1100...1100....1010110010..11010010.1100110010...1100.................................................
............................................1100...1100...1010.110100..10110100.1100110100...1100...............................................
..............................................1100...1100..1010..101100.10.101100.1100101100...1100.............................................
................................................1100...1100.1010..10.110010.10.1100.1101001100...1100...........................................
..................................................1100...11001010..10..11010010..1100.1011001100...1100.........................................
....................................................1100...11010100.10...10110100..110010.11001100...1100.......................................
......................................................1100...1010110010...10.101100..110100.11001100...1100.....................................
........................................................1100..1010.110100..10.10.1100..101100.11001100...1100...................................
..........................................................1100.1010..101100.10.10..1100.10.1100.11001100...1100.................................
............................................................11001010..10.110010.10...110010..1100.11001100...1100...............................
..............................................................11010100.10..11010010....110100..1100.11001100...1100.............................
................................................................1010110010...10110100....101100..1100.11001100...1100...........................
.................................................................1010.110100..10.101100...10.1100..1100.11001100...1100.........................
..................................................................1010..101100.10.10.1100..10..1100..1100.11001100...1100.......................
...................................................................1010..10.110010.10..1100.10...1100..1100.11001100...1100.....................
....................................................................1010..10..11010010...110010....1100..1100.11001100...1100...................
.....................................................................1010..10...10110100...110100....1100..1100.11001100...1100.................
......................................................................1010..10...10.101100...101100....1100..1100.11001100...1100...............
.......................................................................1010..10...10.10.1100..10.1100....1100..1100.11001100...1100.............
........................................................................1010..10...10.10..1100.10..1100....1100..1100.11001100...1100...........
.........................................................................1010..10...10.10...110010...1100....1100..1100.11001100...1100.........
..........................................................................1010..10...10.10....110100...1100....1100..1100.11001100...1100.......
...........................................................................1010..10...10.10.....101100...1100....1100..1100.11001100...1100.....
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Walk representation
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1η
ξ
Excursion: configuration between two successive records
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Pitman transformation
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Pitman transformation
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Pitman transformation
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Identifying solitons (Takahashi-Satsuma)
Call runs the segments induced by broken lines in the walk representation
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1η
ξ
Explore runs from left to right. If a run has length k 6 length of thenext run, then its k boxes and the first k boxes of the next run form ak-soliton. Remove these sites and start again exploring from the left.
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Identifying solitons (Takahashi-Satsuma)
Call runs the segments induced by broken lines in the walk representation
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1η
ξ
Explore runs from left to right. If a run has length k 6 length of thenext run, then its k boxes and the first k boxes of the next run form ak-soliton. Remove these sites and start again exploring from the left.
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Identifying solitons (Takahashi-Satsuma)
Call runs the segments induced by broken lines in the walk representation
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1η
ξ
Explore runs from left to right. If a run has length k 6 length of thenext run, then its k boxes and the first k boxes of the next run form ak-soliton. Remove these sites and start again exploring from the left.
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Following solitons
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Following solitons
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Following solitons
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Speed
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Asymptotic speed of solitons
η shift-ergodic and T -invariant, ρk := number of k-solitons per excursion
Theorem. There exists deterministic v = (vk)k>1 such that, a.s.,
limt→∞
Position of γtt
= vk
for every k-soliton γ of η. The sequence v solves
vk = k +∑m<k
2mρm(vk − vm)−∑m>k
2kρm(vm − vk).
and can be computed explicitly from (ρk)k>1.
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Examples
10111101000010...10111101000010...10111101000010...010...10111101000010...10111101000010...10111101000101000010...10111101000010...10111101000010...10111.10111101000010...10111101000010...10111101000010..001010...101111010000...10111101000010...1011110100110101000010...101111000010...10111101000010...1011..10111101000010...10111101000010...10111101000010.0001010...101111010000...10111101000010...1011110101110101000010...101111000010...10111101000010...101...10111101000010...10111101000010...10111101000010
v1 = 13 , v4 = 6
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Examples
10111101000010...10111101000010...10111101000010...010...10111101000010...10111101000010...10111101000101000010...10111101000010...10111101000010...10111.10111101000010...10111101000010...10111101000010..001010...101111010000...10111101000010...1011110100110101000010...101111000010...10111101000010...1011..10111101000010...10111101000010...10111101000010.0001010...101111010000...10111101000010...1011110101110101000010...101111000010...10111101000010...101...10111101000010...10111101000010...10111101000010
v1 = 13 , v4 = 6
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Examples
1011110110001000.1011110110001000.1011110110001000.01000010.1110111001000010.1110111001000010.11101110101111010000100.1101111010000100.1101111010000100.101000.1011110110001000.1011110110001000.1011110110010111001000010.1110111001000010.1110111001000010.110100.1101111010000100.1101111010000100.11011110100010110001000.1011110110001000.1011110110001000.10111010.1110111001000010.1110111001000010.11101110010001010000100.1101111010000100.1101111010000100.110111.1011110110001000.1011110110001000.1011110110001000
v1 = 19 , v5 = 23
3
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Examples
1011110110001000.1011110110001000.1011110110001000.01000010.1110111001000010.1110111001000010.11101110101111010000100.1101111010000100.1101111010000100.101000.1011110110001000.1011110110001000.1011110110010111001000010.1110111001000010.1110111001000010.110100.1101111010000100.1101111010000100.11011110100010110001000.1011110110001000.1011110110001000.10111010.1110111001000010.1110111001000010.11101110010001010000100.1101111010000100.1101111010000100.110111.1011110110001000.1011110110001000.1011110110001000
v1 = 19 , v5 = 23
3
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Simulations
2000× 140, i.i.d. with density λ = .15
red straight lines are deterministic and computed by the theorem
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Simulations
2000× 140, i.i.d. with density λ = .25
red straight lines are deterministic and computed by the theorem
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Equation for speeds
Why
vk = k +∑m<k
2mρm(vk − vm)−∑m>k
2kρm(vm − vk) ?
Isolated k-solitons have speed k
When a k-soliton encounters an m-soliton:
• it advances 2m extra units if m < k or
• it is delayed by 2 time steps if m > k.
ρm|vk− vm| is the frequency of such encounters as seen from a k-soliton.
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Slots and components
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Nesting Solitons
.11111110000000..1111100000.111000.10..................................................
........11111110000000111110000111001000...............................................
...............11111110000011110001101110000000........................................
......................11111000011100100011111110000000.................................
...........................111100011011100000..11111110000000..........................
...............................11100100.1111100000....11111110000000...................
..................................11011000...1111100000......11111110000000............
....................................10.111000.....1111100000........11111110000000.....
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Nesting Solitons
.7{}..5[].3().1..................................................
........7{5[3(1)]}...............................................
...............7{5[3(1)]}........................................
......................5[3(1)]7{}.................................
...........................5[3(1)]..7{}..........................
...............................3(1).5[]....7{}...................
..................................3(1)...5[]......7{}............
....................................1.3().....5[]........7{}.....
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Slots
3-slots: sites where a 3-soliton can be appended
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Enumerating the k-slots
enumerating 3-slots enumerating 1-slots
Requires an arbitrary:labeling of records
or choice of “Record 0”or walk representation
![Page 40: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/40.jpg)
Enumerating the k-slots
−10
12
3456
enumerating 3-slots
−10
1234 5
6
7
89 10
11121314
enumerating 1-slots
Requires an arbitrary:labeling of records
or choice of “Record 0”or walk representation
![Page 41: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/41.jpg)
Enumerating the k-slots
−10
12
3456
enumerating 3-slots
−10
1234 5
6
7
89 10
11121314
enumerating 1-slots
Requires an arbitrary:labeling of records
or choice of “Record 0”or walk representation
![Page 42: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/42.jpg)
Slot configuration
bb b b b b
b
b
b
b
b
bb
b
b
b
b
b
b
bb
b
b
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b
b
b
bb
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b
b
b
b
b
b
bb b
bbb b
bbb
bb
bb bb
bb bb bb
bb
b
b
bb bb bbbb
bb
bb bb
bb
bb
bb
bb bb bb
bb
bb bb bb
bb
b
b
b
b
b
b
bb
b
b
b
b
b
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bb
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b
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bb
bb
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bb
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bb
b
b
b
b
b
b
bb
b
b
b
b
b
b
bb
b bb b b bb b bb bb b bb b b bb bb bb b bb b b bb123456∞
Sξ
ξ
0
![Page 43: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/43.jpg)
Nested solitonsbb b b b b
b
b
b
b
b
bb
b
b
b
b
b
b
bb
b
b
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b
b
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bb
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b
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b
b
b
b
b
b
b
b
b
bb b
bbb b
bbb
bb
bb bb
bb bb bb
bb
b
b
bb bb bbbb
bb
bb bb
bb
bb
bb
bb bb bb
bb
bb bb bb
bb
b
b
b
b
b
b
bb
b
b
b
b
b
b
bb
b
b
b
b
b
b
bb
bb
b
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b
b
b
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bb
b
b
b
b
b
b
bb
b
b
b
b
b
b
bb
b
b
b
b
b
b
bb
b bb b b bb b bb bb b bb b b bb bb bb b bb b b bb123456∞
Sξ
ξ
0
∞
6
1 2 3 4 5 1 2 3 4 5
4
1 2 3 1 2 3
5
1 2 3 4 1 2 3 4
1 1
Record 0
0-th k-slot for all k2nd 1-slot and 1st 2-slot
4th 4-slot, 2nd 5-slot, etc
11th 1-slot, 8th 2-slot and 5th 3-slot
![Page 44: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/44.jpg)
Soliton components ζ = Mξ ∈ (NZ)N∗
∞
6
1 2 3 4 5 1 2 3 4 5
4
1 2 3 1 2 3
5
1 2 3 4 1 2 3 4
1 1
Record 0
0-th k-slot for all k2nd 1-slot and 1st 2-slot
4th 4-slot, 2nd 5-slot, etc
11th 1-slot, 8th 2-slot and 5th 3-slot
1.......................................................1.....................................................1........................................................................................................................................................................1........1...................................
![Page 45: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/45.jpg)
Invariant measures
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Invariant measures with specified components
Theorem. Let ζ = (ζk ∈ NZ)k>1 be independent fields with shift-invariant law for each k, and whose average density αk satisfy∑
k
kαk <∞.
Then there is a random η whose law µ is shift-invariant and such thatMη is distributed as ζ. This law µ is unique, and it is T -invariant.
If moreover ζk is an i.i.d. field for each k, then µ is also shift-ergodic.
![Page 47: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/47.jpg)
Counter-examples
Invariant ergodic state whose components are not independent
10111101000010...10111101000010...10111101000010...010...10111101000010...10111101000010...10111101000101000010...10111101000010...10111101000010...10111.10111101000010...10111101000010...10111101000010..
Invariant non-ergodic state with independent ergodic components
1011110110001000.1011110110001000.1011110110001000.01000010.1110111001000010.1110111001000010.11101110101111010000100.1101111010000100.1101111010000100.101000.1011110110001000.1011110110001000.10111101100
![Page 48: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/48.jpg)
Evolution of components
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Evolution of components
.11111110000000..1111100000.111000.10..................................................
........11111110000000111110000111001000...............................................
...............11111110000011110001101110000000........................................
......................11111000011100100011111110000000.................................
...........................111100011011100000..11111110000000..........................
...............................11100100.1111100000....11111110000000...................
..................................11011000...1111100000......11111110000000............
....................................10.111000.....1111100000........11111110000000.....
![Page 50: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/50.jpg)
Evolution of components
.7{}..5[].3().1..................................................
........7{5[3(1)]}...............................................
...............7{5[3(1)]}........................................
......................5[3(1)]7{}.................................
...........................5[3(1)]..7{}..........................
...............................3(1).5[]....7{}...................
..................................3(1)...5[]......7{}............
....................................1.3().....5[]........7{}.....
![Page 51: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/51.jpg)
Evolution of components
Ball configuration.11111110000000..1111100000.111000.10..................................................
Fuzzy representation.7{}..5[].3().1..................................................
Components1.................................................................................................................1....................................................................................................................1........................................................................................................................1.........................
![Page 52: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/52.jpg)
Evolution of components
Ball configuration........11111110000000111110000111001000...............................................
Fuzzy representation........7{5[3(1)]}...............................................
Components.......1...............................................................................................................1..................................................................................................................1......................................................................................................................1........................
![Page 53: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/53.jpg)
Evolution of components
Ball configuration...............11111110000011110001101110000000........................................
Fuzzy representation...............7{5[3(1)]}........................................
Components..............1.............................................................................................................1................................................................................................................1....................................................................................................................1.......................
![Page 54: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/54.jpg)
Evolution of components
Ball configuration......................11111000011100100011111110000000.................................
Fuzzy representation......................5[3(1)]7{}.................................
Components.....................1...........................................................................................................1..............................................................................................................1..................................................................................................................1......................
![Page 55: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/55.jpg)
Evolution of components
Ball configuration...........................111100011011100000..11111110000000..........................
Fuzzy representation...........................5[3(1)]..7{}..........................
Components............................1.........................................................................................................1............................................................................................................1................................................................................................................1.....................
![Page 56: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/56.jpg)
Evolution of components
Ball configuration...............................11100100.1111100000....11111110000000...................
Fuzzy representation...............................3(1).5[]....7{}...................
Components...................................1.......................................................................................................1..........................................................................................................1..............................................................................................................1....................
![Page 57: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/57.jpg)
Evolution of components
Ball configuration..................................11011000...1111100000......11111110000000............
Fuzzy representation..................................3(1)...5[]......7{}............
Components..........................................1.....................................................................................................1........................................................................................................1............................................................................................................1...................
![Page 58: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/58.jpg)
Evolution of components
Ball configuration....................................10.111000.....1111100000........11111110000000.....
Fuzzy representation....................................1.3().....5[]........7{}.....
Components.................................................1...................................................................................................1......................................................................................................1..........................................................................................................1..................
![Page 59: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/59.jpg)
Evolution of components
Ball configuration....................................10.111000.....1111100000........11111110000000.....
Fuzzy representation....................................1.3().....5[]........7{}.....
Components1......1......1......1......1......1......1......1................................................................1....1....1....1....1....1....1....1.................................................................................1..1..1..1..1..1..1..1...................................................................................................11111111..................
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Dynamics of components is a hierarchical translation
Theorem. Components of T̂ tξ are shifts components of ξ:
MkT̂tξ = θo
tk(ξ)+ktMkξ
More concisely:T̂ = M−1 ΘM
J tm(ξ) := Flow of m-solitons through Record 0
otk(ξ) :=∑m>k 2(m− k)J tm(ξ)
So otk is determined by (Mmξ)m>k
![Page 61: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/61.jpg)
Reconstruction
![Page 62: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/62.jpg)
Reconstruction algorithm ξ = M−1ζ
......................................................
......................................................
......................................................
......................................................1.......................................................1.....................................................1........................................................................................................................................................................1........1...................................ζ
↓
bb b b b b
b
b
b
b
b
bb
b
b
b
b
b
b
bb
b
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bb
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b
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b
b
b
b
b
b
b
b
bb b
bbb b
bbb
bb
bb bb
bb bb bb
bb
b
b
bb bb bbbb
bb
bb bb
bb
bb
bb
bb bb bb
bb
bb bb bb
bb
b
b
b
b
b
b
bb
b
b
b
b
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b
bb
b
b
b
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bb
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bb
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bb
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b
b
b
b
b
bb
b
b
b
b
b
b
bb
b bb b b bb b bb bb b bb b b bb bb bb b bb b b bb123456∞
Sξ
ξ
0
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Reconstruction algorithm ξ = M−1ζ
0 5 10 15
54321 0 2 1 0 0 0 0 1 1 1 0 0 2 0 0 1 0 0 0 0 0 0 0 1 0 2
0 0 01 2 0 10 0 0 0 00 1 10 0 00 0 1 00 0 00
...
k = i
... ... ... ...
-5
x
ξ
![Page 64: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/64.jpg)
Reconstruction of ε0
∞
∞
4
1 2 3 1 2 3
∞
4
1 2 3 1 2 3
3
1 2 1 2
∞
4
1 2 3 1 2 3
3
1 2 1 2
2
1 1
2
1 1
∞ε0
4
1 2 3 1 2 3
3
1 2 1 2
2
1 1
2
1 1
1
1 1 1 1
![Page 65: Soliton decomposition of the Box-Ball Systemmate.dm.uba.ar/~leorolla/bbs-slides.pdf · 2020-05-19 · Identifying solitons (Takahashi-Satsuma) Call runs the segments induced by broken](https://reader033.vdocuments.net/reader033/viewer/2022043003/5f8374134ddcba3bc907df43/html5/thumbnails/65.jpg)
Reconstruction of the other excursions:
Reconstruction of ε0
∞ε1
2
1 1
1
1 1
∞ε−1
3
1 2 1 2
2
1 1
11
1 1
∞ε−2
empty
x
ξ