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A finite Toda representation of the box-ball system with box capacity

A connection between the finite ultradiscrete Toda lattice and the box-ball system is extended to the case where each box has own capacity and a carrier has a capacity parameter depending on time. In order to consider this connection, new carrier rules"size limit for solitons"and"recovery of balls", and a concept"expansion map"are introduced. A particular solution to the extended system of a special case is also presented.


Introduction
The box-ball system (BBS), proposed by Takahashi and Satsuma [9], is one of the most important cellular automata obtained from discrete integrable systems through a limiting procedure called ultradiscretization [12]. It is well known that the time evolution of the original BBS is determined by the ultradiscrete KdV (u-KdV) equation: where U (t) n ∈ {0, 1} denotes the number of balls in the nth box at time t. The equation (1) is an ultradiscrete analogue of the discrete KdV (d-KdV) equation, and consequently the original BBS has ultradiscrete soliton solutions. It is also known that the nonautonomous discrete KP (nd-KP) equation with 2-reduction condition yields a time evolution equation of the BBS with three extensions [1]: box capacity [10], carrier capacity [8], and kind of balls [11].
In this paper, we consider another type of time evolution equations of the BBS first presented by Nagai et al [5]. They discovered that the ultradiscrete Toda (u-Toda) equation on a (non-periodic) finite lattice which we refer to as the finite u-Toda lattice simply, determines the time evolution of the original BBS. In this equation, the dependent variables Q (t) n and E (t) n denote the size of the nth soliton at time t and the size of the nth empty block at time t, respectively. Figure 1 shows an example of the connection between the finite u-Toda lattice and the BBS. We call this representation finite Toda representation of the BBS. The correspondence between the u-KdV equation and the finite u-Toda equation via the BBS is similar to the Euler-Lagrange correspondence of cellular automaton [3]. This terminology comes from hydrodynamics; the dependent variables of the Euler representation denote the number of particles at each point and the ones of the Lagrange representation denote the position of each particle. According to these definitions, we use the following terms in this paper: The u-KdV equation (1) and the finite u-Toda lattice (2) gives the Euler representation and the Lagrange representation of the original BBS, respectively. Additionally, if we know the start position of the first soliton, we can calculate the start positions of all solitons and empty blocks from the values of the variables of the finite Toda representation. In other words, the finite Toda representation and the Lagrange representation can be transformed to each other. It has not been clarified by now why these two different ultradiscrete equations with different type boundary conditions describe the same original BBS. Moreover, the finite Toda representation of the extended BBSs are not studied sufficiently. Among the three extensions for the BBS, two types of the finite Toda representations have been clarified: Tokihiro et al [11] discussed the case of with several kinds of balls using the finite ultradiscrete hungry Toda lattice; Tsujimoto and the author [2] showed that the finite nonautonomous ultradiscrete Toda (nu-Toda) lattice determines the time evolution of the BBS with a carrier. The main purpose of this paper is to discuss the remaining case; we derive a finite Toda representation of the BBS with box capacity. For this purpose, we use a map from a state of the BBS to a binary sequence, which we call "expansion map". By using the expansion map, we can define the size of solitons in the BBS with box capacity for all time t, especially for interacting solitons. We also consider the finite Toda representation of the BBS with both box capacity and carrier capacity using new carrier rules "size limit for solitons" and "recovery of balls". Furthermore, we present a particular solution for the fixed box capacity case.
The outline of this paper is the following. In section 2, we recall the derivation of the Euler representation of the BBS with a carrier from the 2-reduced nd-KP equation. In addition, we introduce new carrier rules "size limit for solitons" and "recovery of balls" which are used to derive the finite Toda representation of the BBS with a carrier [2] and play an important role in section 4. In section 3, we recall the finite Toda representation of the original BBS and give some remarks. In section 4, we discuss the finite Toda representation of the BBS with variable box capacity. First we discuss the case of variable box capacity ∆ n and no restricted carrier capacity. After that, we discuss the case of both box capacity and carrier capacity are variable. In section 5, we give a particular solution to the finite Toda representation of the BBS with fixed box capacity. In section 6, we give concluding remarks.

Euler representation of the BBS with a carrier
The nd-KP equation is given by [13] (a It is shown that the N -soliton solution to the nd-KP equation (3) is presented by where ξ i , p i and q i , i = 0, 1, . . . , N − 1, are some constants. Now we impose the 2reduction condition with respect to the variable k, that is f k+2,t n = f k,t n and c k+2 = c k for all k ∈ Z, and set a n = 1 + δ n , b t = −µ t , c 0 = 1 and c 1 = 0. Then the nd-KP equation (3) reduces to the forms and an N -soliton solution to the reduced equations is given by holds. For positivity, we choose the parameters as 0 ≤ δ n ≤ 1 and 0 ≤ µ t ≤ 1 for all n, t ∈ Z. When the values of the dependent variables are all positive for all n, t ∈ Z, we can ultradiscretize the equations (7): putting u n /ǫ , δ n = e −∆n/ǫ , µ t = e −Mt/ǫ into (7) and taking a limit ǫ → +0, we obtain the ultradiscrete system where ∆ n , M t ≥ 0 for all n, t ∈ Z. Note that we have used the fundamental formula for the ultradiscretization: An N -soliton solution to the ultradiscrete system (8) is obtained as follows. Let us take the constants p i , i = 0, 1, . . . , N − 1, to satisfy the condition 0 < p i < 1. Putting f k,t n = e −F k,t n /ǫ , h k,t i,n = e −H k,t i,n /ǫ , p i = e −Pi/ǫ , ξ i = e −Ξi/ǫ , w i,j = e −Wi,j /ǫ into (5) and (6), and taking a limit ǫ → +0, we obtain and P i ≥ 0, i = 0, 1, . . . , N − 1. Let us introduce the time evolution rule of the BBS with the nth box capacity ∆ n and the carrier capacity M t+1 from time t to t + 1. We consider the time evolution rule from time t to t + 1 as the composition of size limit process and recovery process.
(i) Size limit process: the carrier of balls moves from left (n = −∞) to right (n = +∞). When the carrier passes each box, the carrier gets all balls in the box; and if the number of balls exceeds the carrier capacity M t+1 , the excess balls are removed from the system. At the same time, the carrier puts the balls which the carrier holds into the box as many as possible. (ii) Recovery process: after the size limit process, all the removed balls are recovered to the boxes in which the balls were. Figure 2 illustrates these rules. Suppose that the dependent variables U • Z (t+1) n ∈ {0, 1, . . . , M t+1 }: the number of balls in the carrier arriving at the nth box in the size limit process from time t to t + 1.
Then the equations (8) give the time evolution rule: the equations (8a) and (8b) define the size limit process and, since gives the number of removed balls by the size limit at the nth box, the equation (8c) defines the recovery process Remark 1. Eliminating the variable U (t+1) n from the equations (8a) and (8b), we have the relation also holds. Substituting (10) into (9), we obtain the equation where we have used the formula The equation (11) has the same form as of the time evolution equation of the "BBS with a carrier" presented by Takahashi and Matsukidaira [8].
If we choose M t+1 = +∞, then (11) yields which is the nonautonomous u-KdV equation. In addition, (8b) yields and comparing this relation with (8c), we have the relation U

Finite Toda representation of the original BBS
We recall the relation between the finite u-Toda lattice and the original BBS. The bilinear form of the discrete Toda lattice is given by Let us introduce the dependent variables Then (13) yields the equation and the identities n = e −D (t) n /ǫ , and taking a limit ǫ → +0, we obtain the u-Toda lattice Furthermore, we impose the terminating condition for discussing the finite Toda representation: where N is a positive integer, which denotes the number of solitons in the original BBS.
Let the variables Q Then the next theorem gives a fundamental result on the connection between the finite u-Toda lattice and the BBS.
Theorem 1 (Nagai et al [5]). The finite u-Toda lattice (15) determines the time evolution of the original BBS.
Remark 2. Conventionally, the d-Toda lattice (qd-type) has been written in the form which we cannot ultradiscretize directly due to "negative problem". The equations (16) are called Rutishauser's qd algorithm in numerical algorithms [6]. On the other hand, the equations (14) are called dqd algorithm. The dqd algorithm is the subtraction-free form of the qd algorithm and computes matrix eigenvalues or singular values more accurately than the qd algorithm. Suppose the finite lattice condition e Then they could ultradiscretize the finite Toda lattice: On the other hand, by introducing an auxiliary variable n+1 , we can ultradiscretize the finite d-Toda lattice directly without the negative problem and obtain the finite u-Toda lattice of the dqd form (15). From the viewpoint of the BBS, the variable D (t+1) n denotes the number of balls in the carrier. Therefore, the finite u-Toda lattice of the dqd form (15) is important to consider the finite Toda representation of the BBS with a carrier. Remark 3. Here we remark on the Lagrange representation of the BBS, which is also a terminology from hydrodynamics as the Euler representation; the dependent variables of the Lagrange representation denote the position of solitons. Let the variables X (t) n and Y (t) n denote the start position of the nth soliton and the one of the nth empty block at time t, respectively (see Figure 4). Then the Lagrange representation of the BBS is given by [4] Relations between these variables and the variables of the finite u-Toda lattice (17) are given by We can readily show that (18) and (19) . In other words, the finite u-Toda lattice (17) and the equation X 0 , which is obtained from (18a) and (19), uniquely determine the time evolution of the BBS. In previous studies, the finite Toda representation is considered only for the BBS with box capacity 1. In this section, we extend the finite Toda representation to the case in which each box has own capacity ∆ n . First we consider the case of carrier capacity M t = +∞. The Euler representation of this case is given by (12). We first define the size of solitons and the one of empty blocks for the BBS with variable box capacity ∆ n at any time t. For this purpose, we refer to the work of Takahashi and Satsuma [10]. They analyzed the BBS with the fixed box capacity ∆ using a map from a state of box capacity ∆ to a binary sequence. We generalize this map for the case of variable box capacity ∆ n .
Suppose that a state of the Euler representation (12) n ∈ {0, 1, . . . , ∆ n } is given. We assume that, for simplicity, U Note that the relation U We refer the jth number V (t) j in the binary sequence as the jth segment. By using this map, the nth box is expanded to the block composed from the s n th to (s n + ∆ n − 1)th segments. We call this map expansion map from a state of the BBS with variable box capacity ∆ n to a binary sequence. Figure 5. Example of the expansion map from a state of the BBS with variable box capacity ∆n to a binary sequence. In the binary sequence, a block between two separators '|' corresponds to an original box. Figure 6. Example of the expansion map for the BBS with variable box capacity ∆n. The carrier capacity parameter is chosen as Mt = +∞ for all t ∈ Z. Figures 5 and 6 show examples of the expansion map. As shown in Figure 6, the expansion map enables us to define the size of the nth soliton Q We note that, from (20a), Q  Let us consider the time evolution of the BBS with box capacity ∆ n from time t to t + 1. Assume that Q (t+1) j , j = 0, 1, . . . , n − 1, and E (t+1) j , j = 1, 2, . . . , n − 1, are given (see Figure 7). Let m be the index of the box which contains the leftmost segment of the (n + 1)th empty block at time t. Then the capacity of the mth box ∆ m is equal to Λ (t) n+1 by definition. Moreover, the relation k denotes the number of balls in the kth box at time t, also holds by definition. Hence, we can calculate the quantity U (t+1) m by the nu-KdV equation (12): . Then we obtain the relation where we have used the identity − min(−A, −B) = max (A, B). This relation implies that the term max(0, Λ ) denotes the size of interspace inserted between the nth soliton at time t and the nth one at time t + 1.
Once we notice the role of the terms max(0, Λ , we can now clarify the meaning of the equations (20a) and (20b). Since the term E in (20a) denotes the difference between the size of the inserted space and the one of the (n + 1)th empty block, Q (t+1) n should be determined by (20a). Similarly, E (t+1) n should be determined by (20b). It is also true for n = 0 and 1, then the proof is completed by induction.
Next, we construct the finite Toda representation of the BBS with both box capacity and carrier capacity from two time evolution maps: the size limit map and the recovery map. This is a similar as for the construction of the Euler representation explained in section 2. Figure 8 shows an example.
In the next theorem, we use the following notations: n : the size of the nth soliton and the one of the nth empty block at time t, respectively; : the size of the nth soliton and the one of the nth empty block after the size limit process from time t to t + 1; : some quantities which will be explained in the proof of the next theorem in detail; n : the capacity of the box which contains the leftmost segment of the nth soliton and the one of the nth empty block at time t, respectively. Figure 8. Example of the expansion map for the BBS with box capacity ∆n and carrier capacity Mt = 6 for t > 0. This is obtained from the example in Figure 3 via the expansion map. n , and carrier capacity M t+1 is given by where the carrier capacity M t+1 must be chosen to satisfy the condition K   If M t+1 = +∞, then (21c) and (21d) reduce to C . Hence, in this case, the ultradiscrete system (21) reduces to the system (20). We can therefore say that the system (21) is a generalization of the system (20).
Proof. Let us show that the equations (21a)-(21d) describe the size limit process and (21e)-(21f) describe the recovery process.
First, we consider the size limit process. The equations (21a) and (21b) have the same forms as of (20a) and (20b). Thus we shall investigate the variables holds from the assumption. We should note that, in this case, the number of balls which the carrier holds is zero temporarily before getting Q m should hold (see Figure 9). Now ∆ m = K (t) n by definition. Hence we can write (21c) and (21d) as C Size limit to M t+1 balls D  Table 1.
Thus, together with the proof of Theorem 3, it is proved that (21a)-(21d) describe the size limit process by induction.
Furthermore, since the number of balls removed by the size limit process are given by (D , we obtain the equations of the recovery process which lead to the equations (21e) and (21f), and the proof is completed.

Particular solution for the fixed box capacity case
In this section, we discuss a particular solution to the ultradiscrete system (21) with a special condition: all boxes have constant capacity ∆.
Let us consider the bilinear equations τ 0,t+1 n τ 1,t−1 n = δτ 0,t−1 n+1 τ 1,t+1 n−1 + τ 0,t n τ 1,t n , (23a) δτ 0,t n τ 1,t n = (δ − µ t )τ 0,t n τ 1,t n + µ t τ 0,t+1 where δ is a constant and µ t is a parameter depending on t. We introduce the dependent variables Then (23d) yields the relation Further, (23a)-(23c) yield the equations and the identities hold. In addition, we impose the finite lattice condition In the bilinear equations (23), this condition implies We assume that the constant δ and the parameter µ t satisfy the condition 0 < µ t < δ for all t ∈ Z. Then, putting q and taking a limit ǫ → +0, we obtain the ultradiscrete system (21) with the condition K (t) n = Λ (t) n = ∆ ≤ M t+1 for all n, t ∈ Z. The following theorem is proved by using a determinant identity called the Plücker relation.

Concluding remarks
In this paper, we have derived the finite Toda representation of the BBS with box capacity by introducing the expansion map from a state of the BBS to a binary sequence. Furthermore, we have given a particular solution for the fixed box capacity case. Hence we can say that the ultradiscrete system (21) is integrable if the parameters K (t) n and Λ (t) n are chosen as constants. Since there is a connection between the ultradiscrete system (21) and the BBS with variable box capacity which is integrable, we expect that the ultradiscrete system (21) of the variable box capacity case is also integrable and a discrete system derived through the inverseultradiscretization has determinant solutions. This problem is left for future research.
In the proof of Theorem 3, the variables C (t+1) n and D (t+1) n have played important roles; these variables denote the number of balls which the carrier has. Moreover, these variables correspond to the variables which are introduced to remove subtractions in the discrete equations (see Remark 2). This result gives us a guideline for ultradiscretization of Toda-type integrable systems and making connections between these systems and BBSs.
In 2000, Spiridonov and Zhedanov [7] proposed a Toda-type nonautonomous integrable system called R II chain, which is derived as the compatibility conditions of spectral transformations for some biorthogonal rational functions. By using techniques developed in this paper, we will be able to ultradiscretize the R II chain and consider a corresponding BBS.