Another generalization of the box–ball system with many kinds of balls
Kazuki Maeda (Kwansei Gakuin University, Japan)

Email: kmaeda@kmaeda.net

𝑄10 𝑄20 𝑄30 𝐸01 𝑄11 𝑄21 𝑄31 𝐸11 𝑄12 𝑄22 𝑄32

𝑄10 𝑄20 𝑄30 𝐸01 𝑄11 𝑄21 𝑄31 𝐸11 𝑄12 𝑄22 𝑄32

𝑡 = 0: .1222333...11223...133............................ 1 3 3 3 2 2 1 3 1 0 2

𝑡 = 0: .1222333...11223...133............................ 1 3 3 3 2 2 1 3 1 0 2𝑡 = 1: ..1122233....11233..223........................... 2 3 2 4 2 1 2 2 0 2 1
𝑡 = 2: ....1122233....12233112........................... 2 3 2 4 1 2 2 0 2 1 0𝑡 = 3: ......1122233...1122..1333........................ 2 3 2 3 2 2 0 2 1 0 3𝑡 = 4: ........1122233...1133.2223....................... 2 3 2 3 2 0 2 1 0 3 1𝑡 = 5: ..........1122233...22311123...................... 2 3 2 3 0 2 1 0 3 1 1𝑡 = 6: ............1122233.112...12333................... 2 3 2 1 2 1 0 3 1 1 3𝑡 = 7: ..............112223..1333.12223.................. 2 3 1 2 1 0 3 1 1 3 1𝑡 = 8: ................112233.2223.11123................. 2 2 2 1 0 3 1 1 3 1 1𝑡 = 9: ..................1122311223...12333.............. 2 2 1 0 2 2 1 3 1 1 3𝑡 =10: ....................112..122333.122233............ 2 1 0 2 1 2 3 1 1 3 2
𝑡 =11: ......................133.122223.111223........... 1 0 2 1 1 4 1 1 3 2 1𝑡 =12: .......................223.111123...122333........ 0 2 1 1 4 1 1 3 1 2 3𝑡 =13: .......................112.....12333.1222233...... 2 1 0 5 1 1 3 1 1 4 2𝑡 =14: .........................133....12223.1112223..... 1 0 2 4 1 3 1 1 3 3 1𝑡 =15: ..........................223....11123...1122333.. 0 2 1 4 3 1 1 3 2 2 3𝑡 = 1: .......1222333..1122..1333........................ 1 3 3 2 2 2 0 2 1 0 3
𝑡 = 2: .............1222333112...12333................... 1 3 3 0 2 1 0 3 1 1 3𝑡 = 3: ...................122311223...123333............. 1 2 1 0 2 2 1 3 1 1 4𝑡 = 4: .......................223..11123...1223333....... 0 2 1 2 3 1 1 3 1 2 4𝑡 = 5: ..........................223....11123....1223333. 0 2 1 4 3 1 1 4 1 2 4Figure 1

Example of the known BBS with many kinds of balls and limited carrier
capacity (𝑀 = 3, carrier capacity: 6)

Figure 2

Example of the novel BBS with many kinds of balls and limited numberof rewrites (𝑀 = 3, 𝑆 (𝑡) = 2 for all 𝑡)

1 Introduction

A generalization of the BBS by Hatayama et al. [1] is known (Ex.: Figure 1)

the box with a ball whose index is the smallest among the indexes of the balls,
whose indexes are larger than 𝑘, in the carrier (ex.: if 𝑘 = 1 and 𝑀 = 3, thenthe order is 2 < 3 < 0 < 1).

We propose another generalization of the BBS [2] (Ex.: Figure 2)⋙ RuleThe novel rule (Ex.: Figure 2):

There is an array of infinite boxes. Several boxes hold one ball.

– A machine moves from left to right. The carrier has a state called“the numbers of rewrites” (𝑉 (0) , 𝑉 (1) , … , 𝑉 (𝑀) ). Initial state at time 𝑡 is
(𝑉 (0) , 𝑉 (1) , … , 𝑉 (𝑀) ) = (0, 𝑆 (𝑡) , … , 𝑆 (𝑡) ).
– When the machine passes each box, if the index of the ball in the box is 𝑘: if𝑉 (𝑘) ≥ 1 then rewrites the index 𝑘 as 𝑘 − 1 and sets the state 𝑉 (𝑘) ← 𝑉 (𝑘) − 1,𝑉 (𝑘−1) ← 𝑉 (𝑘−1) + 1; if 𝑉 (𝑘) = 0 then does nothing.

Each ball has an index (from 1 to 𝑀). We regard empty boxes as balls with index
0 (periods . indicates 0 in the figures). Note that we regard the indexes cyclically:𝑘 ≡ 𝑘 mod (𝑀 + 1).
The known rule (Ex.: Figure 1):– A carrier of balls moves from left to right. The carrier can hold a finite numberof balls.
– When the carrier passes each box, the carrier exchanges the ball with index 𝑘 in

There are two types of the time evolution equations corresponding to the novel BBS
as follows.

2 Reduced nonautonomous ultradiscrete KP lattice𝑈𝑛(𝑘,𝑡+1) = 𝑈𝑛(𝑘,𝑡) − 𝑋𝑛(𝑘,𝑡) + 𝑋𝑛(𝑘+1,𝑡) ,𝑋𝑛(𝑘,𝑡) = min(𝑈𝑛(𝑘,𝑡) , 𝑉𝑛(𝑘,𝑡) ),(𝑘−1,𝑡)(𝑘−1,𝑡+1)
𝑋𝑛(𝑘,𝑡) = 𝐹𝑛(𝑘−1,𝑡) − 𝐹𝑛+1+ 𝐹𝑛+1− 𝐹𝑛(𝑘−1,𝑡+1) ,(𝑘,𝑡)
𝑉𝑛+1= 𝑉𝑛(𝑘,𝑡) − 𝑋𝑛(𝑘,𝑡) + 𝑋𝑛(𝑘+1,𝑡) ,wherewhere 𝑈𝑛(𝑘+𝑀+1,𝑡) = 𝑈𝑛(𝑘,𝑡) and 𝑉𝑛(𝑘+𝑀+1,𝑡) = 𝑉𝑛(𝑘,𝑡) for all 𝑘, 𝑡 and 𝑛, with the boundary
conditions for 𝑛 → −∞:⎧⎧1if
𝑘=0,if 𝑘 = 0,{{0(𝑘,𝑡)(𝑘,𝑡)𝑈𝑛=⎨
𝑉𝑛= ⎨ (𝑡){{if 𝑘 = 1, 2, … , 𝑀.
⎩0 if 𝑘 = 1, 2, … , 𝑀,
⎩𝑆𝑚−1
⎛⎞
⎜
⎟
⎛
⎞
(𝑘,𝑡)
(𝑘,𝑡)
̃⎜
⎜
⎟
𝐹𝑛= min ⎜0,min∑ 𝛹𝑟𝑖 ,𝑛 (𝑖) ⎟,⎟0≤𝑟0 <𝑟1 <⋯<𝑟𝑚−1 ≤𝑁−1
⎝ 𝑖=0⎠⎠
𝑚=1,2,…,𝑁⎝(𝑘′ +𝑙+𝑖) mod 𝑀′+𝑙+𝑖𝑘⎛(𝑡) ⎞(𝑘,𝑡)⎜̃̃⌋ + 𝑖) 𝑍𝑟 + 𝑆𝑡−𝑙 ⎟𝛹𝑟,𝑛 (𝑖) ≔ 𝛩𝑟 + min ⎜∑𝜁𝑟,𝑗 + (⌊⎟𝑙=0,1,…,𝑡𝑀𝑗=1⎝⎠− 𝑛 min(𝑍𝑟 , 1).

𝑈𝑛(𝑘,𝑡) ∈ {0, 1}: the number of balls with index 𝑘 in the 𝑛th box at time 𝑡;𝑉𝑛(𝑘,𝑡) : “the number of rewrites of index 𝑘” of the machine arriving at the 𝑛th box
at time 𝑡;𝑋𝑛(𝑘,𝑡) : 1 if 𝑈𝑛(𝑘,𝑡) = 1 and 𝑉𝑛(𝑘,𝑡) ≥ 1, otherwise 0.
An 𝑁-soliton solution is derived from Baker–Akhiezer functions via reduction andultradiscretization procedure:(𝑀,𝑡) − 𝐹 (𝑀,𝑡) + 𝐹 (0,𝑡) − 𝐹 (0,𝑡) + 1 if 𝑘 = 0,⎧𝐹{ 𝑛𝑛𝑛+1𝑛+1(𝑘,𝑡)𝑈𝑛= ⎨ (𝑘−1,𝑡)(𝑘−1,𝑡)(𝑘,𝑡)(𝑘,𝑡){𝐹−𝐹+𝐹−𝐹if 𝑘 = 1, 2, … , 𝑀,𝑛⎩ 𝑛𝑛+1𝑛+1(𝑀,𝑡) − 𝐹 (𝑀,𝑡+1) + 𝐹 (0,𝑡+1) − 𝐹 (0,𝑡)⎧𝐹if 𝑘 = 0,{𝑛𝑛𝑛𝑛(𝑘,𝑡)𝑉𝑛= ⎨ (𝑘−1,𝑡)(𝑘−1,𝑡+1) + 𝐹 (𝑘,𝑡+1) − 𝐹 (𝑘,𝑡) + 𝑆 (𝑡) if 𝑘 = 1, 2, … , 𝑀,{𝐹−𝐹𝑛𝑛𝑛⎩ 𝑛Here, 𝑘′ ≔ 𝑘 mod (𝑀 + 1), 𝑍𝑟 ≥ 0 and 𝛩𝑟 are the parameters determine the sizeand phase of the soliton, respectively, 𝜁𝑟,1 , 𝜁𝑟,2 , … , 𝜁𝑟,𝑀 are nonnegative parameters𝑀satisfying ∑𝑗=1 𝜁𝑟,𝑗 = 𝑍𝑟 , and⎧0{{𝑖−1{(𝑡)⎛̃⎜∑ 𝑆 (𝑗𝑙 ) ⎞⎟𝑆𝑖 ≔ ⎨
min{0≤𝑗0 <𝑗1 <⋯<𝑗𝑖−1 ≤𝑡−1 ⎝ 𝑙=0⎠{{+∞⎩

if 𝑖 = 0,if 𝑖 = 1, 2, … , 𝑡,otherwise.

3 Reduced nonautonomous ultradiscrete 2D Toda latticeOr, so-called nonautonomous ultradiscrete hungry Toda lattice:(𝑘,𝑡) , 𝐴(𝑘,𝑡) ),𝑄̃ (𝑘,𝑡)=min(𝑄𝑘 = 1, … , 𝑀,𝑛𝑛𝑛(1,𝑡) , 𝐸 (1,𝑡) ),𝑄̃ (𝑀+1,𝑡)=min(𝐵𝑛𝑛𝑛(𝑘,𝑡) − 𝑄̃ (𝑘,𝑡) + 𝑄̃ (𝑘+1,𝑡) ,𝑄(𝑘,𝑡+1)=𝑄
𝑘 = 1, … , 𝑀,𝑛𝑛𝑛𝑛(1,𝑡)(1,𝑡+1)(1,𝑡)(𝑀+1,𝑡)̃̃𝐸𝑛= 𝐸𝑛 − 𝑄𝑛+ 𝑄𝑛+1 ,(𝑘,𝑡)(𝑘,𝑡) + 𝑄̃ (𝑘+1,𝑡) ,̃
𝐴𝑛+1 = 𝐴(𝑘,𝑡)−𝑄
𝑘 = 1, … , 𝑀,𝑛𝑛𝑛(1,𝑡)̃𝐵 (1,𝑡) = 𝐵𝑛(1,𝑡) − 𝑄̃ (𝑀+1,𝑡)+𝑄,𝑛𝑛+1𝑛+1with the boundary conditions(𝑘,𝑡)
𝐴0 = 𝑆 (𝑡) ,(1,𝑡)(1,𝑡)𝐵0 = min(𝑄0 , 𝑆 (𝑡) ),block of empty boxes at time 𝑡;
Particular solutions are derived from biorthogonal polynomials as(𝑘,𝑡) − 𝑇 (𝑘,𝑡) + 𝑇 (𝑘+1,𝑡) − 𝑇 (𝑘+1,𝑡) ,𝑄(𝑘,𝑡)=𝑇𝑛𝑛𝑛𝑛+1𝑛+1
(1,𝑡)
(1,𝑡)
(𝑀+1,𝑡)𝐸𝑛(1,𝑡) = 𝑇𝑛+2
− 𝑇𝑛+1+ 𝑇𝑛(𝑀+1,𝑡) − 𝑇𝑛+1,(𝑘,𝑡) − 𝑇 (𝑘+1,𝑡) + 𝑇 (𝑘+1,𝑡+1) − 𝑇 (𝑘,𝑡+1) + 𝑆 (𝑡) ,𝐴(𝑘,𝑡)=𝑇𝑛𝑛𝑛𝑛𝑛(1,𝑡)(1,𝑡+1)
𝐵𝑛(1,𝑡) = 𝑇𝑛(𝑀+1,𝑡) − 𝑇𝑛+1+ 𝑇𝑛+1− 𝑇𝑛(𝑀+1,𝑡+1) ,where𝑛−1

𝑘 = 1, … , 𝑀,𝑇𝑛(𝑘,𝑡) =(1,𝑡)𝐸𝑁 = +∞.

𝑘 + 𝑀𝑗 + 𝑐𝑗(𝑘+𝑐𝑗 )min( ∑ (𝑊𝑟𝑗+𝑍𝑟𝑗 )0≤𝑟0 <𝑟1 <⋯<𝑟𝑛−1 ≤𝑁−1
𝑀
𝑗=00≤𝑐0 <𝑐1 <⋯<𝑐𝑛−1 ≤𝑡+𝑛−1(𝑡)̂+𝑆

𝑁: the number of solitons;
𝑄(𝑘,𝑡): the number of balls with index 𝑘 in the 𝑛th soliton at time 𝑡;𝑛𝐸𝑛(1,𝑡) : the number of the 𝑛th block of empty boxes at time 𝑡;
𝐴(𝑘,𝑡): “the number of rewrites of index 𝑘” of the machine arriving at the 𝑛th𝑛soliton at time 𝑡;𝐵𝑛(1,𝑡) : “the number of rewrites of index 0” of the machine arriving at the 𝑛thReferences(𝑡−𝑐0 ,𝑡+1−𝑐1 ,…,𝑡+𝑛−1−𝑐𝑛−1 )′𝑡−1),

⎛⎞
(𝑗)
⎜⎟
≔
min∑
𝑦𝑆
⎜
⎟.
′
𝑗(𝜆0 ,𝜆1 ,…,𝜆𝑛−1 )𝑌⎝𝑗=0⎠Here, 𝑊𝑟(𝑘) are constants satisfying 𝑊𝑟(𝑘+𝑀) = 𝑊𝑟(𝑘) , min indicates the minimum𝑌value over all semistandard Young tableaux 𝑌 of the partition (𝜆0 , 𝜆1 , … , 𝜆𝑛−1 )′ , and𝑦0 , 𝑦1 , … , 𝑦𝑛−1 are the weights of 𝑌.
(𝑡)
̂𝑆[1] G. Hatayama, K. Hikami, R. Inoue, A. Kuniba, T. Takagi and T. Tokihiro, The 𝐴(1)𝑀 automata related to crystals of symmetric tensors, J. Math. Phys. 42 (2001), 274–308.
[2] K. Maeda, Another generalization of the box–ball system with many kinds of balls, J. Integrable Syst. 3 (2018), xyy007.