Figure 6 - uploaded by Tetsuji Tokihiro
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An example of a block. The '1' pointed at by vertical arrows are the components of the largest soliton in the block.
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We investigate periodic box–ball systems (PBBSs) with several kinds of balls and box capacity greater than or equal to one. Conserved quantities of the PBBSs are constructed from those of the nonautonomous discrete KP (ndKP) equation using the Lax representation of the ndKP equation.
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Citations
... In section 2, we derive the path description of characteristic polynomial of particular matrices. In section 3, we briefly summarize the results of Ref. [9], which we will use in the subsequent sections. In section 4, we treat the ndKP equation which corresponds to the GPBBS. ...
... limit of the constrained ndKP equation $(3\mathrm{A})$ with the periodic boundary condition coincides with the time evolution equation of the GPBBS .[9] we derived the Lax representation for the ndKP equation when it has period $N$ in the spatial variable $n$ . ...
The present article is a review on the conserved quantities of periodic box-ball systems (PBBS) with arbitrary kinds of balls and box capacity greater than or equal to one. By introducing the notion of nonintersecting paths on the two dimensional array of boxes, we give a combinatorial formula for the conserved quantities of the generalized PBBS using these paths.
... In section 2, we derive the path description of characteristic polynomial of particular matrices. In section 3, we briefly summarize the results of Ref. [9], which we will use in the subsequent sections. In section 4, we treat the ndKP equation which corresponds to the GPBBS. ...
... The upper bound of the summation over $j$ is $\ovalbox{\tt\small REJECT}\frac{N-k}{N}(M+1)\ovalbox{\tt\small REJECT}$ where $[ $(-1)^{j(N-k-1)}\mu^{j}$ $1. \leq..d_{1}<d_{2}<\sum_{<d_{N-k}\leq N}\cdots(F_{1,..\prime}P_{N-k})\sum_{\in P^{(j)_{(d_{1\prime}..,d_{N-k})}}}\prod_{i=1}^{N-k}\prod_{n=1}^{N}\prod_{m=0}^{M}\xi_{n,m}(P_{i})$ where $\xi_{n,m}$ and $\mathcal{P}^{(j)}(d_{1}$ , . . . , $d_{N-k})$ are defined in (2.4) and (2.5) Then we have the following theorem Theorem 3.1 ([9]) The time evolution of the GPBBS is described by a ultradiscrete equation: ...
... To take the ultradiscrete limit, we put $U_{n}^{s}=e^{u_{n}^{s}/\epsilon}$ , $K_{n}^{s}=e^{\kappa_{n}^{s}/\epsilon}$ , $1/\tilde{\delta}_{n+1}=$ $e^{\theta_{n}/\epsilon}$ . Then we found Theorem 3.2 ([9]) The ultradiscrete limit of the constrained ndKP equation $(3\mathrm{A})$ with the periodic boundary condition coincides with the time evolution equation of the GPBBS (3.2), $\square$ ...
We investigate conserved quantities of periodic box-ball systems (PBBS) with arbitrary kinds of balls and box capacity greater than or equal to 1. We introduce the notion of nonintersecting paths on the two dimensional array of boxes, and give a combinatorial formula for the conserved quantities of the generalized PBBS using these paths.
A box-ball system with more than one kind of ball is obtained by the generalized periodic discrete Toda equation (pd Toda equation). We present an algebraic geometric study of the periodic Toda equation. The time evolution of the pd Toda equation is linearized on the Picard group of an algebraic variety, and theta function solutions are obtained.
We investigate the fundamental cycle of a periodic box–ball system (PBBS) from a relation between the PBBS and a solvable lattice model. We show that the fundamental cycle of the PBBS is obtained from eigenvalues of the transfer matrix of the solvable lattice model.
The (M,K)-reduced non-autonomous discrete KP equation is linearised on the Picard group of an algebraic curve. As an application, we construct theta function solutions to the initial value problem of some special discrete KP equation. Comment: 20 pages
