Mixed soliton solutions for the ultradiscrete BKP equation and its reduction

Abstract

The soliton solution of the ultradiscrete BKP equation is obtained, via ultradiscretization of the generalized discrete BKP equation. It is shown that these solutions consist of three different kinds of soliton solutions for the ultradiscrete BKP equation. We also derive the soliton solution of the B-type box and ball system by taking the reduction of the ultradiscrete BKP equation and clarify the origin of the two types of soliton solutions of the B-type box and ball system.

Appendix A: Transformation

We shall derive (4) from

$$\begin{aligned} {\bar{f}}(p, q, r) =\sum _{k_1=1}^2\sum _{k_2=1}^2\ldots \sum _{k_N=1}^2 \prod _{1\le i<j\le N} cf(t_i^{(k_j)}, t_j^{(k_i)}) \prod _{i=1}^N c_i^{(k_i)} \phi (t_i^{(k_i)}), \end{aligned}$$

(A.1)

which is a soliton solution of (1) given in [9]. Here $t_i^{(k_i)}$ and $c_i^{(k_i)}$ are arbitrary parameters, $\phi (t)$ and cf(t, s) are defined by (5) and (8). Using the gauge transformation, that is, by multiplying (A.1) with

$$\begin{aligned} \frac{1}{\prod _{1\le i<j\le N }cf(t_i^{(2)}, t_j^{(2)})\prod _{i=1} ^Nc_i^{(2)}\phi (t_i^{(2)})}, \end{aligned}$$

we have another form of soliton solution,

$$\begin{aligned} f(p, q, r) =\sum _{k_1=1}^2\sum _{k_2=1}^2\ldots \sum _{k_N=1}^2 \prod _{1\le i<j\le N} \frac{cf(t_i^{(k_j)}, t_j^{(k_i)})}{cf(t_i^{(2)}, t_j^{(2)}) } \prod _{i=1}^N \frac{c_i^{(k_i)} \phi (t_i^{(k_i)})}{c_i^{(2)} \phi (t_i^{(2)})}. \end{aligned}$$

(A.2)

Note that the term associated with

$$\begin{aligned} k_i={\left\{ \begin{array}{ll} 1 &{}\quad (i\in I:=\{i_1, i_2,\ldots , i_l \}) \\ 2 &{}\quad (i\in I^c:=[N]- \{i_1, i_2,\ldots , i_l \}) \end{array}\right. } \end{aligned}$$

in (A.2) is expressed by

$$\begin{aligned} \prod _{i\in I }\frac{c_i^{(1)}}{c_i^{(2)}}\frac{\phi (t_i^{(1)})}{\phi (t_i^{(2)})}\left( \prod _{\substack {1\le i<j\le N \\ i, j\in I }} \frac{cf(t_i^{(1)}, t_j^{(1)})}{cf(t_i^{(2)}, t_j^{(2)}) }\prod _{\substack {1\le i<j\le N \\ i\in I , j\in I^c} } \frac{cf(t_i^{(2)}, t_j^{(1)})}{cf(t_i^{(2)}, t_j^{(2)}) }\prod _{\substack {1\le i<j\le N \\ i\in I^c, j\in I} } \frac{cf(t_i^{(1)}, t_j^{(2)})}{cf(t_i^{(2)}, t_j^{(2)}) } \right) . \end{aligned}$$

(A.3)

If we replace free parameters $c_i$ as

$$\begin{aligned} \frac{c_i^{(1)}}{c_i^{(2)}} \rightarrow c_i\prod _{\substack{1\le j\le N \\ j\not =i} } \frac{ cf(t_i^{(2)}, t_j^{(2)})}{ cf(t_i^{(2)}, t_j^{(1)})}, \end{aligned}$$

then (A.3) is reduced to

$$\begin{aligned} \prod _{i\in I }c_i\frac{\phi (t_i^{(1)})}{\phi (t_i^{(2)})}\prod _{\substack {i,j\in I \\ i<j } } b_{ij} \end{aligned}$$

(A.4)

since $cf(t, s)=-cf(s, t)$ and

$$\begin{aligned} \prod _{\substack {i\in I \\ j\not =i } }b_{ij} = \prod _{\substack {i, j\in I \\ i<j } }b_{ij} \prod _{\substack {i, j\in I \\ i<j } }b_{ji} \prod _{\substack {i\in I, j \in I^c \\ i<j } }b_{ij} \prod _{\substack {j\in I, i \in I^c \\ i<j} } b_{ji} \end{aligned}$$

hold. Therefore (A.2) corresponds to (4).