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Ultradiscrete permanent solution to the ultradiscrete Kadomtsev–Petviashvili equation

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Abstract

We propose an ultradiscrete permanent solution to the ultradiscrete Kadomtsev–Petviashvili (KP) equation. The ultradiscrete permanent is an ultradiscrete analogue of the usual permanent. The elements on this ultradiscrete permanent solution are required some additional relations other than the ultradiscrete dispersion relation. We confirm the solution satisfying these relations and propose some explicit examples of the solution.

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Acknowledgements

The author is grateful to Professor Daisuke Takahashi for helpful advice. The author also grateful Keisuke Mizuki for valuable discussions.

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Correspondence to Hidetomo Nagai.

Appendices

Appendix A: Proof of (15)

In this appendix, we prove Proposition 1. From the definition, \(\mathrm{up}[B\otimes C]\) is expressed by

$$\begin{aligned} \mathrm{up}[B \otimes C] = \mathrm{up}\begin{bmatrix} {\displaystyle \max _{1\le k\le M}}(b_{1k}+c_{k1})&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{1k}+c_{kN}) \\ \vdots&\ddots&\vdots \\ {\displaystyle \max _{1\le k\le M}}(b_{Nk}+c_{k1})&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{Nk}+c_{kN}) \end{bmatrix}. \end{aligned}$$
(44)

Applying (12) and (13) to the first column, (44) is expanded as the maximum of M UPs as below.

$$\begin{aligned}&\mathrm{up}[B\otimes C]\nonumber \\&\quad =\max \left( \mathrm{up}\begin{bmatrix} b_{11}+c_{11}&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{1k}+c_{kN}) \\ \vdots&\ddots&\vdots \\ b_{N1}+c_{11}&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{Nk}+c_{kN}) \end{bmatrix}, \right. \nonumber \\&\qquad \mathrm{up}\begin{bmatrix} b_{12}+c_{21}&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{1k}+c_{kN}) \\ \vdots&\ddots&\vdots \\ b_{N2}+c_{21}&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{Nk}+c_{kN}) \end{bmatrix}, \ \dots , \nonumber \\&\qquad \left. \mathrm{up}\begin{bmatrix} b_{1M}+c_{M1}&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{1k}+c_{kN}) \\ \vdots&\ddots&\vdots \\ b_{NM}+c_{M1}&\ldots&{\displaystyle \max _{1\le k\le M}}(b_{Nk}+c_{kN}) \end{bmatrix}\right) \nonumber \\&\quad = \max _{1\le k_1\le M} \left( c_{k_11}+\mathrm{up}\begin{bmatrix} b_{1k_1}&\ldots&\max _{1\le k\le M}(b_{1k}+c_{kN}) \\ \vdots&\ddots&\vdots \\ b_{Nk_1}&\ldots&\max _{1\le k\le M}(b_{Nk}+c_{kN}) \end{bmatrix}\right) . \end{aligned}$$
(45)

Applying similar procedure to the other columns, we obtain

$$\begin{aligned} \mathrm{up}[B\otimes C]= \max _{1\le k_1, k_2, \dots , k_N\le M} \left( \sum _{1\le i\le N} c_{k_ii} +\mathrm{up}\begin{bmatrix} b_{1k_1}&\ldots&b_{1k_N} \\ \vdots&\ddots&\vdots \\ b_{Nk_1}&\ldots&b_{Nk_N} \end{bmatrix}\right) . \end{aligned}$$
(46)

It is equivalent to

$$\begin{aligned} \mathrm{up}[B\otimes C]=\max _{1\le j_1\le \dots \le j_N\le M} \left( \max _{\pi '}\left( \sum _{1\le i\le N} c_{\pi ' _ii} +\mathrm{up}\begin{bmatrix} b_{1\pi '_1}&\ldots&b_{1\pi '_N} \\ \vdots&\ddots&\vdots \\ b_{N\pi '_1}&\ldots&b_{N\pi '_N} \end{bmatrix}\right) \right) , \end{aligned}$$
(47)

where \(\pi '=(\pi '_1, \pi '_2, \dots , \pi '_N)\) is a set of all possible permutations of \(\{j_1, j_2, \dots , j_N \}\). In particular, the UP of the matrix whose columns are exchanged is the same as original one from (11). Therefore, we obtain

$$\begin{aligned} \mathrm{up}[B\otimes C]= & {} \max _{1\le j_1\le \dots \le j_N\le M} \left( \max _{\pi ' } \sum _{1\le i\le N} c_{\pi '_ii} +\mathrm{up}\begin{bmatrix} b_{1j_1}&\ldots&b_{1j_N} \\ \vdots&\ddots&\vdots \\ b_{Nj_1}&\ldots&b_{Nj_N} \end{bmatrix}\right) \nonumber \\= & {} \max _{1\le j_1\le \dots \le j_N\le M} \left( \mathrm{up}[C]^{j_1\dots j_N}_{1\dots N} +\mathrm{up}[B]^{1\dots N}_{j_1\dots j_N} \right) . \end{aligned}$$
(48)

Thus (15) holds.

Appendix B: Proof of (16)

In this appendix, we express

$$\begin{aligned} \varvec{a} = \begin{bmatrix} a_1\\ a_2 \\ \vdots \\ a_N\end{bmatrix}, \quad \varvec{b} = \begin{bmatrix} b_1\\ b_2 \\ \vdots \\ b_N\end{bmatrix}, \quad \varvec{c} = \begin{bmatrix} c_1\\ c_2 \\ \vdots \\ c_N\end{bmatrix}, \end{aligned}$$

respectively. Then \(\mathrm{up}[D_{l , m, n}]+\mathrm{up}[D_{l' , m', n'}]\) is expressed by

$$\begin{aligned}&\mathrm{up}[D_{l , m, n}] +\mathrm{up}[D_{l' , m' , n' }] \nonumber \\&\quad = \max _{\pi , \pi ' }\Bigl ( a_{\pi _1}+\dots + a_{\pi _l}+b_{\pi _{l+1}}+\dots +b_{\pi _{l+m}}+ c_{\pi _{l+m+1}}+\dots + c_{\pi _N} \nonumber \\&\qquad +\, a_{\pi '_1}+\dots + a_{\pi ' _{l' }}+b_{\pi ' _{l' +1}}+\dots +b_{\pi ' _{l' +m'}}+ c_{\pi '_{l' +m' +1}}+\dots + c_{\pi ' _N}\Bigr ), \end{aligned}$$
(49)

namely, \(\mathrm{up}[D_{l , m, n}]+\mathrm{up}[D_{l' , m', n'}]\) can be expressed by

$$\begin{aligned}&a_{\pi _1}+\dots + a_{\pi _l}+b_{\pi _{l+1}}+\dots +b_{\pi _{l+m}}+ c_{\pi _{l+m+1}}+\dots + c_{\pi _N} \nonumber \\&\quad +\, a_{\pi '_1}+\dots + a_{\pi ' _{l' }}+b_{\pi ' _{l' +1}}+\dots +b_{\pi ' _{l' +m'}}+ c_{\pi '_{l' +m' +1}}+\dots + c_{\pi ' _N} \end{aligned}$$
(50)

for certain permutations \(\pi \) and \(\pi '\). In particular, due to \(m>m'\), there exists j such that

$$\begin{aligned}&j\in \{ \pi _{l+1 }, \pi _{l +2 }, \dots , \pi _{l+m }\}\quad \text {and }\\&\quad j\in \{ \pi '_1 , \pi '_2 , \dots , \pi '_{l'}\} \cup \{ \pi '_{l'+m'+1} , \pi '_{l'+m'+2} , \dots , \pi '_N\}. \end{aligned}$$

First, let us consider in the case there exists \(j_0\) such that

$$\begin{aligned} j_0\in \{ \pi _{l+1 }, \pi _{l +2 }, \dots , \pi _{l+m }\}\quad \text {and } \quad j_0\in \{ \pi '_1 , \pi '_2 , \dots , \pi '_{l'}\}. \end{aligned}$$
(51)

Then \(\mathrm{up}[D_{l, m, n}] +\mathrm{up}[D_{l' , m' , n' }]\) can be expanded as

$$\begin{aligned} \mathrm{up}[D_{l, m, n}] +\mathrm{up}[D_{l' , m' , n' }] =\mathrm{up}\left[ D_{l, m, n} \begin{bmatrix} j_0 \\ l+1 \end{bmatrix}\right] +b_{j_0} + \mathrm{up}\left[ D_{l' , m' , n'} \begin{bmatrix} j_0 \\ 1 \end{bmatrix}\right] +a_{j_0} \end{aligned}$$
(52)

where \(D \begin{bmatrix} j \\ k \end{bmatrix}\) denotes the \((N-1)\times (N-1)\) matrix obtained by eliminating the jth row and the kth column from D. On the other hand, \(\mathrm{up}[D_{l+1 , m-1 , n}] +\mathrm{up}[D_{l'-1 , m'+1 , n' }]\) can be evaluated as

$$\begin{aligned}&\mathrm{up}[D_{l+1 , m-1 , n}] +\mathrm{up}[D_{l'-1 , m'+1 , n' }] \nonumber \\&\quad \ge \mathrm{up}\left[ D_{l+1 , m-1 , n}\begin{bmatrix} j_0 \\ 1 \end{bmatrix}\right] +a_{j_0} +\mathrm{up}\left[ D_{l'-1 , m'+1 , n' }\begin{bmatrix} j_0 \\ l' \end{bmatrix}\right] +b_{j_0}. \end{aligned}$$
(53)

From (52) and (53), we obtain

$$\begin{aligned} \mathrm{up}[D_{l+1 , m-1 , n}] +\mathrm{up}[D_{l'-1 , m'+1 , n' }]\ge \mathrm{up}[D_{l , m , n}] +\mathrm{up}[D_{l' , m' , n' }] \end{aligned}$$
(54)

since

$$\begin{aligned} \mathrm{up}\left[ D_{l, m, n} \begin{bmatrix} j_0 \\ l+1 \end{bmatrix}\right]= & {} \mathrm{up}\left[ D_{l+1 , m-1 , n}\begin{bmatrix} j_0 \\ 1 \end{bmatrix}\right] ,\nonumber \\ \mathrm{up}\left[ D_{l' , m' , n'} \begin{bmatrix} j_0 \\ 1 \end{bmatrix}\right]= & {} \mathrm{up}\left[ D_{l'-1 , m'+1 , n' }\begin{bmatrix} j_0 \\ l' \end{bmatrix}\right] \end{aligned}$$
(55)

hold.

Next we consider the case there is no \(j_0\) such that (51). Then there exists \(j_1\) such that

$$\begin{aligned} j_1\in \{ \pi _{l+1 }, \pi _{l +2 }, \dots , \pi _{l+m }\}\quad \text {and } \quad j_1\in \{ \pi '_{l'+m'+1} , \pi '_{l'+m'+2} , \dots , \pi '_N\} . \end{aligned}$$

In addition, due to \(l<l'\), there also exists \(j_2\) such that

$$\begin{aligned} j_2\in \{ \pi _{l +m+1} , \pi _{l+m+2} , \dots , \pi _N \} \quad \text {and } \quad j_2\in \{ \pi '_1 , \pi '_2 , \dots , \pi '_{l'} \}. \end{aligned}$$

Thus, \(\mathrm{up}[D_{l, m, n}] +\mathrm{up}[D_{l' , m' , n' }]\) can be expanded as

$$\begin{aligned}&\mathrm{up}[D_{l, m, n}] +\mathrm{up}[D_{l' , m' , n' }] \nonumber \\&\quad = \mathrm{up}\left[ D_{l, m , n} \begin{bmatrix} j_1&j_2 \\ l+1&l +m+1\end{bmatrix}\right] +b_{j_1}+c_{j_2} \nonumber \\&\qquad +\, \mathrm{up}\left[ D_{l' , m' , n'} \begin{bmatrix}j_1&j_2 \\ l'+m'+1&1 \end{bmatrix}\right] +c_{j_1}+a_{j_2} . \end{aligned}$$
(56)

On the other hand \(\mathrm{up}[D_{l+1 , m-1 , n}] +\mathrm{up}[D_{l'-1 , m'+1 , n' }]\) can be evaluated as

$$\begin{aligned}&\mathrm{up}[D_{l+1 , m-1 , n}] +\mathrm{up}[D_{l'-1 , m'+1 , n' }] \nonumber \\&\quad \ge \mathrm{up}\left[ D_{l+1 , m-1 , n }\begin{bmatrix} j_1&j_2 \\ l+m+1&1 \end{bmatrix}\right] +c_{j_1}+a_{j_2} \nonumber \\&\qquad +\,\mathrm{up}\left[ D_{l'-1 , m'+1 , n' }\begin{bmatrix} j_1&j_2 \\ l'&l'+m'+1 \end{bmatrix}\right] +b_{j_1}+c_{j_2}. \end{aligned}$$
(57)

Thus we obtain \(\mathrm{up}[D_{l+1 , m-1 , n}] +\mathrm{up}[D_{l'-1 , m'+1 , n' }]\ge \mathrm{up}[D_{l , m , n}] +\mathrm{up}[D_{l' , m' , n' }]\) since

$$\begin{aligned} \mathrm{up}\left[ D_{l, m , n} \begin{bmatrix} j_1&j_2 \\ l+1&l +m+1\end{bmatrix}\right]= & {} \mathrm{up}\left[ D_{l+1 , m-1 , n }\begin{bmatrix} j_1&j_2 \\ l+m+1&1 \end{bmatrix}\right] , \nonumber \\ \mathrm{up}\left[ D_{l' , m' , n'} \begin{bmatrix}j_1&j_2 \\ l'+m'+1&1 \end{bmatrix}\right]= & {} \mathrm{up}\left[ D_{l'-1 , m'+1 , n' }\begin{bmatrix} j_1&j_2 \\ l'&l'+m'+1 \end{bmatrix}\right] \end{aligned}$$
(58)

hold.

Appendix C: Proof of (35)

We prove (35) in the case of \(M=1, 2, 3\) and any integer N. We can assume

$$\begin{aligned} \varphi _i(s) = \max _{1\le j\le M}( c_{ij}+p_js) \end{aligned}$$
(59)

and

$$\begin{aligned} p_1\le p_2\le \dots \le p_M \end{aligned}$$
(60)

without a loss of generality. One can prove (35) in the case of \(M=1\). In the case of \(M=2\), we add \(-(c_{i1}+c_{i2})/2\) to each ith row and \(-(p_1s+p_2s)/2\) to each column \(\varvec{\varphi }(s)\) for all the UP in (35). In other words, for the property (12), we add \(-\sum _{i=1}^N(c_{i1}+c_{i2})- \sum _{i=1}^N (p_1+p_2)i- (p_1+p_2)(N+1-k_1-k_2-k_3)/2\) to both sides in (35). Then each \(\varphi _i(s)\) is reduced as

$$\begin{aligned} \varphi _i(s)-\frac{c_{i1}+c_{i2}+p_1s+p_2s}{2} = \frac{1}{2}|c_{i1}-c_{i2}+(p_1-p_2)s|. \end{aligned}$$

Thus, (35) can be rewritten as

$$\begin{aligned}&\mathrm{up}\left[ \varvec{\varphi }(0) \cdots \widehat{\varvec{\varphi }(k_2)}\cdots \varvec{\varphi }(N)\right] + \mathrm{up}\left[ \varvec{\varphi }(0)\cdots \widehat{\varvec{\varphi }(k_1)}\cdots \widehat{\varvec{\varphi }(k_3)}\cdots \varvec{\varphi }(N+1)\right] \nonumber \\&\quad \ge \mathrm{up}\left[ \varvec{\varphi }(0) \cdots \widehat{\varvec{\varphi }(k_1)}\cdots \varvec{\varphi }(N)\right] + \mathrm{up}\left[ \varvec{\varphi }(0)\cdots \widehat{\varvec{\varphi }(k_2)}\cdots \widehat{\varvec{\varphi }(k_3)}\cdots \varvec{\varphi }(N+1)\right] ,\nonumber \\ \end{aligned}$$
(61)

where

$$\begin{aligned} {\varvec{\varphi }(s)}= \begin{pmatrix} |c'_{1}+p s| \\ |c'_{2}+p s| \\ \dots \\ |c'_{N}+p s| \end{pmatrix} \end{aligned}$$
(62)

with arbitrary parameters \(c'_i\) and \(p'\). This inequality has been proved in [6]. Hence (35) in the case of \(M=2\) holds.

We consider in the case of \(M=3\). Using (15), \(\mathrm{up}[\varvec{\varphi }(0) \cdots \widehat{\varvec{\varphi }(k_2)}\cdots \varvec{\varphi }(N)]\) can be expanded as

$$\begin{aligned}&\mathrm{up}\left[ \varvec{\varphi }(0) \cdots \widehat{\varvec{\varphi }(k_2)}\cdots \varvec{\varphi }(N)\right] \\&\quad = \mathrm{up}\begin{bmatrix} \begin{bmatrix} c_{11}&c_{12}&c_{13} \\ c_{21}&c_{22}&c_{23} \\ \vdots&\vdots&\vdots \\ c_{N1}&c_{N2}&c_{N3} \end{bmatrix} \otimes \begin{bmatrix} 0&p_1&2p_1&\ldots&\widehat{k_2p_1}&\ldots&Np_1\\ 0&p_2&2p_2&\ldots&\widehat{k_2p_2}&\ldots&Np_2\\ 0&p_3&2p_3&\ldots&\widehat{k_2p_3}&\ldots&Np_3 \end{bmatrix} \end{bmatrix}\\&\quad = \max _{1\le i_1\le \dots \le i_N\le 3} \left( \mathrm{up}\begin{bmatrix} c_{1i_1}&c_{1i_2}&\ldots&c_{1i_N} \\ c_{2i_1}&c_{2i_2}&\ldots&c_{2i_N} \\ \vdots&\vdots&\ddots&\vdots \\ c_{Ni_1}&c_{Ni_2}&\ldots&c_{Ni_N} \end{bmatrix} + \mathrm{up}\begin{bmatrix} 0&p_{i_1}&2p_{i_1}&\ldots&\widehat{k_2p_{i_1}}&\ldots&Np_{i_1}\\ 0&p_{i_2}&2p_{i_2}&\ldots&\widehat{k_2p_{i_2}}&\ldots&Np_{i_2}\\ \vdots&\vdots&\vdots&\ddots&\vdots&\ddots&\vdots \\ 0&p_{i_N}&2p_{i_N}&\ldots&\widehat{k_2p_{i_N}}&\ldots&Np_{i_N} \end{bmatrix} \right) \\&\quad = \max _{1\le i_1\le \dots \le i_N\le 3}\left( \mathrm{up}[C_{i_1i_2\dots i_N}]+ \sum _{l=1}^{k_2} (l-1) p_{i_l}+\sum _{l=k_2+1}^{N} l p_{i_l}\right) , \end{aligned}$$

where \( C_{i_1i_2\dots i_N}\) denotes

$$\begin{aligned} C_{i_1i_2\dots i_N}= \begin{bmatrix} c_{1i_1}&c_{1i_2}&\ldots&c_{1i_N} \\ c_{2i_1}&c_{2i_2}&\ldots&c_{2i_N} \\ \vdots&\vdots&\ddots&\vdots \\ c_{Ni_1}&c_{Ni_2}&\ldots&c_{Ni_N} \end{bmatrix}. \end{aligned}$$

By similar procedure, we obtain

$$\begin{aligned}&\mathrm{up}[\varvec{\varphi }(0) \cdots \widehat{\varvec{\varphi }(k_2)}\cdots \varvec{\varphi }(N)]+ \mathrm{up}[\varvec{\varphi }(0)\cdots \widehat{\varvec{\varphi }(k_1)}\cdots \widehat{\varvec{\varphi }(k_3)}\cdots \varvec{\varphi }(N+1)]\nonumber \\&\quad = \max _{I, J}\left( \mathrm{up}[C_{i_1i_2\dots i_N}]+\mathrm{up}[C_{j_1j_2\dots j_N}]+\sum _{l=1}^N l p_{i_l}\right. \nonumber \\&\qquad \left. + \sum _{l=1}^N lp_{j_l}-\sum _{l=1}^{k_2} p_{i_l}-\sum _{l=1}^{k_1} p_{j_l} +\sum _{l=k_3}^N p_{j_l}\right) ,\end{aligned}$$
(63)
$$\begin{aligned}&\mathrm{up}[\varvec{\varphi }(0) \cdots \widehat{\varvec{\varphi }(k_1)}\cdots \varvec{\varphi }(N)]+ \mathrm{up}[\varvec{\varphi }(0)\cdots \widehat{\varvec{\varphi }(k_2)}\cdots \widehat{\varvec{\varphi }(k_3)}\cdots \varvec{\varphi }(N+1)]\nonumber \\&\quad =\max _{I', J'}\left( \mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] +\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] +\sum _{l=1}^N l p_{i'_l}\right. \nonumber \\&\qquad \left. + \sum _{l=1}^N lp_{j'_l}-\sum _{l=1}^{k_1} p_{i'_l}-\sum _{l=1}^{k_2} p_{j'_l} +\sum _{l=k_3}^N p_{j'_l}\right) . \end{aligned}$$
(64)

Here we denote \(\max _{1\le i_1\le \dots \le i_N\le 3, 1\le j_1\le \dots \le j_N\le 3}\) as \(\max _{I, J}\) and \(\sum _{l=m}^n\) is defined as 0 when \(m>n\). Let us consider the argument of RHS of (64):

$$\begin{aligned} \mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] +\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] + \sum _{l=1}^N l p_{i'_l}+ \sum _{l=1}^N lp_{j'_l}-\sum _{l=1}^{k_1} p_{i'_l}-\sum _{l=1}^{k_2} p_{j'_l} +\sum _{l=k_3}^N p_{j'_l}. \end{aligned}$$
(65)

Equation (35) in the case of \(M=3\) holds if we show (65) is less than or equal to RHS of (63) for any \(i'_1, i'_2, \dots i'_N\), \(j'_1, j'_2, \dots , j'_N\). Hereafter we fix \(i'_1, i'_2, \dots i'_N\), \(j'_1, j'_2, \dots , j'_N\). First we compare (65) and the arguments of RHS of (63) associated with \(i_l=i'_l\), \(j_i=j'_l\) or \(i_l=j'_l\), \(j_i=i'_l\). Then, we obtain

$$\begin{aligned}&\max _{I, J}\left( \mathrm{up}[C_{i_1i_2\dots i_N}]+\mathrm{up}[C_{j_1j_2\dots j_N}]+\sum _{l=1}^N l p_{i_l}+ \sum _{l=1}^N lp_{j_l}-\sum _{l=1}^{k_2} p_{i_l}-\sum _{l=1}^{k_1} p_{j_l} +\sum _{l=k_3}^N p_{j_l}\right) \nonumber \\&\qquad - \left( \mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] +\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] + \sum _{l=1}^N l p_{i'_l}+ \sum _{l=1}^N lp_{j'_l}-\sum _{l=1}^{k_1} p_{i'_l}-\sum _{l=1}^{k_2} p_{j'_l} +\sum _{l=k_3}^N p_{j'_l}\right) \nonumber \\&\quad \ge \max \left( \mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] +\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] + \sum _{l=1}^N l p_{i'_l}+ \sum _{l=1}^N lp_{j'_l}-\sum _{l=1}^{k_2} p_{i'_l}-\sum _{l=1}^{k_1} p_{j'_l} +\sum _{l=k_3}^N p_{j'_l},\right. \nonumber \\&\mathrm{up}\left. \left[ C_{j'_1j'_2\dots j'_N}\right] +\mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] + \sum _{l=1}^N l p_{j'_l}+ \sum _{l=1}^N lp_{i'_l}-\sum _{l=1}^{k_2} p_{j'_l}-\sum _{l=1}^{k_1} p_{i'_l} +\sum _{l=k_3}^N p_{i'_l}\right) \nonumber \\&\qquad -\left( \mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] +\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] + \sum _{l=1}^N l p_{i'_l}+ \sum _{l=1}^N lp_{j'_l}-\sum _{l=1}^{k_1} p_{i'_l}-\sum _{l=1}^{k_2} p_{j'_l} +\sum _{l=k_3}^N p_{j'_l}\right) \nonumber \\&\quad = \max \left( \sum _{l=k_1+1}^{k_2} (-p_{i'_l}+p_{j'_l}) , \quad \sum _{l=k_3}^N (p_{i'_l} - p_{j'_l})\right) . \end{aligned}$$
(66)

It takes a nonnegative value when

$$\begin{aligned} \sum _{l=k_1+1}^{k_2} (-p_{i'_l}+p_{j'_l})\ge 0 \quad \text {or} \quad \sum _{l=k_3}^N (p_{i'_l} - p_{j'_l})\ge 0 \end{aligned}$$
(67)

holds for any \(0\le k_1<k_2<k_3\le N\). Next let us compare (65) and RHS of(63) in the case

$$\begin{aligned} \sum _{l=k_1+1}^{k_2} (-p_{i'_l}+p_{j'_l})<0 \end{aligned}$$
(68)

and

$$\begin{aligned} \sum _{l=k_3}^N (p_{i'_l} - p_{j'_l})<0 \end{aligned}$$
(69)

hold for certain \(k_1\), \(k_2\) and \(k_3\). We introduce the notations

$$\begin{aligned} i'_l = {\left\{ \begin{array}{ll} 1 &{} \quad (l=1, 2, \dots , \alpha )\\ 2 &{} \quad (l=\alpha +1, \alpha +2, \dots , \beta )\\ 3 &{} \quad (l=\beta +1, \beta +2, \dots , N) \end{array}\right. }, \quad j'_l= {\left\{ \begin{array}{ll} 1 &{} \quad (l=1, 2, \dots , \gamma )\\ 2 &{} \quad (l=\gamma +1, \gamma +2, \dots , \delta )\\ 3 &{} \quad (l=\delta +1, \delta +2, \dots , N) \end{array}\right. }. \end{aligned}$$
(70)

If \(\alpha \ge k_2\), then \(p_{i'_l}\) equals to \(p_1\) when \(l=1, 2, \dots , k_2\) and (68) does not hold. Thus \(\alpha \) should be \(\alpha < k_2\). Similarly, \(\beta \) should be \(\beta \ge k_3\) for (69). Due to the obtained condition \(\alpha<k_2<k_3\le \beta \), \(p_{i'_l}\) takes either \(p_1\) or \(p_2\) when \(l=1, 2, \dots , k_3\). Hence, the condition (68) requires there exists \(l\in \{k_1+1, \dots , k_2 \}\) such that \(p_{i'_l}=2\) and \(p_{j'_l}=1\). Thus \(\gamma \) should be \(\alpha <\gamma \) and \(k_1+1<\gamma \). Similarly, \(\delta \) should be \(\delta <\beta \) for (69). Therefore we obtain \(\alpha< \gamma \le \delta <\beta \), \(\alpha< k_2<k_3\le \beta \le N\) and \(k_1+1<\gamma \). These conditions derive \(i'_{k_2}=i'_{k_3}=2\), \(j'_{k_1+1}=j'_{\alpha }=1\) and \(j'_{\beta }=j'_N =3\) (see Table 1). For these \(i'_l\) and \(j'_l\), we set \(i_l\) and \(j_l\) as

$$\begin{aligned} i_l = {\left\{ \begin{array}{ll} 1 &{} \quad (l=1, 2, \dots , \alpha +1 )\\ 2 &{} \quad (l=\alpha +2, \alpha +3, \dots , \beta )\\ 3 &{} \quad (l=\beta +1, \beta +2, \dots , N) \end{array}\right. }, \quad j_l= {\left\{ \begin{array}{ll} 1 &{} \quad (l=1, 2, \dots , \gamma -1 )\\ 2 &{} \quad (l=\gamma , \gamma +1, \dots , \delta )\\ 3 &{} \quad (l=\delta +1, \delta +2, \dots , N) \end{array}\right. }. \end{aligned}$$
(71)
Table 1 The sets of \(i_l\), \(j_l\), \(i'_l\), \(j'_l\) for (70) and (71)

Subtracting (65) associated with (70) from the argument of RHS of (63) associated with (71), we obtain

$$\begin{aligned}&\mathrm{up}[C_{i_1i_2\dots i_N}]+\mathrm{up}[C_{j_1j_2\dots j_N}]+\sum _{l=1}^N l p_{i_l}+ \sum _{l=1}^N lp_{j_l}-\sum _{l=1}^{k_2} p_{i_l}-\sum _{l=1}^{k_1} p_{j_l} +\sum _{l=k_3}^N p_{j_l}\nonumber \\&\qquad -\, \left( \mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] +\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] + \sum _{l=1}^N l p_{i'_l}+ \sum _{l=1}^N lp_{j'_l}-\sum _{l=1}^{k_1} p_{i'_l}-\sum _{l=1}^{k_2} p_{j'_l} +\sum _{l=k_3}^N p_{j'_l} \right) \nonumber \\&\quad =\mathrm{up}[C_{i_1i_2\dots i_N}]+\mathrm{up}[C_{j_1j_2\dots j_N}]-\mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] -\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] \nonumber \\&\qquad +\, (\gamma -\alpha -1) (p_2-p_1) -\sum _{l=1}^{k_2} (p_{i_l}-p_{j'_l})+\sum _{l=1}^{k_1} (p_{i'_l}-p_{j_l}) +\sum _{l=k_3}^N(p_{j_l}-p_{j'_l})\nonumber \\&\quad \ge \mathrm{up}[C_{i_1i_2\dots i_N}]+\mathrm{up}[C_{j_1j_2\dots j_N}]-\mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] -\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] \nonumber \\&\qquad +\, (\gamma -\alpha -1)(p_2-p_1)-(\gamma -\alpha -1) (p_2-p_1)\nonumber \\&\qquad +\,\sum _{l=1}^{k_1} (p_{i'_l}-p_{j_l}) +\sum _{l=k_3}^N(p_{j_l}-p_{j'_l}) \nonumber \\&\quad = \mathrm{up}[C_{i_1i_2\dots i_N}]+\mathrm{up}[C_{j_1j_2\dots j_N}]-\mathrm{up}\left[ C_{i'_1i'_2\dots i'_N}\right] \nonumber \\&\qquad -\,\mathrm{up}\left[ C_{j'_1j'_2\dots j'_N}\right] +\sum _{l=1}^{k_1} (p_{i'_l}-p_{j_l})+\sum _{l=k_3}^N(p_{j_l}-p_{j'_l}) \end{aligned}$$
(72)

It takes a nonnegative value from Proposition 2 and \(k_1+1<\gamma \). Therefore (35) holds.

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Nagai, H. Ultradiscrete permanent solution to the ultradiscrete Kadomtsev–Petviashvili equation. Japan J. Indust. Appl. Math. 35, 355–372 (2018). https://doi.org/10.1007/s13160-017-0289-1

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