狠狠撸

?
: soliton
2
1965 N. Zabusky M. Kruskal
KdV (KdV: Korteweg-de Vries) 2
-on
solitary wave(-on)
: solitron
Wikipedia

19
1950 60
→
1967
1970
1980
→
1990
...

KdV
?u
?t
+ 6u
?u
?x
+
?3
u
?x3
= 0, (u = u(x, t))
mKdV
?u
?t
+ 6u2 ?u
?x
+
?3
u
?x3
= 0, (u = u(x, t))
KP
?
?x
4
?u
?t
? 6u
?u
?x
?
?3
u
?x3
? 3
?2
u
?y2
= 0, (u = u(x, y, t))
d2
dt2
log(1 + Vn) = Vn+1 ? 2Vn + Vn?1 (Vn = Vn(t))
KdV KP .

KdV
KdV
?u
?t
+ 6u
?u
?x
+
?3
u
?x3
= 0, (u = u(x, t)).
u(x, t) = 2k2
sech2
k(x ? 4k2
t + c), k, c

KdV 2-
KdV
2-
u(x, t) = 2
?2
?x2
log f(x, t)
f(x, t) = 1+eη1
+eη2
+
k1 ? k2
k1 + k2
2
eη1+η2
, ηj(x, t) = kjx?k3
j
t+cj
2-

KP
Kadomtsev-Petviashvili (KP)
?
?x
4
?u
?t
? 6u
?u
?x
?
?3
u
?x3
? 3
?2
u
?y2
= 0, (u = u(x, y, t))
2-
u(x, y, t) = 2
?2
?x2
log τ(x, y, t)
τ(x, y, t) = 1 + eP1x+Q1y+?1t
+ eP2x+Q2y+?2t
+
(p1 ? p2)(q1 ? q2)
(p1 ? q2)(q1 ? p2)
e(P1+P2)x+(Q1+Q2)y+(?+?2)t
,
Pi = pi ? qi, Qi = p2
i
? q2
i
, ?i = p3
i
? q3
i
(i = 1, 2)
pi, qi KP

KP
?
?x
4
?u
?t
? 6u
?u
?x
?
?3
u
?x3
? 3
?2
u
?y2
= 0, (u = u(x, y, t))
u = 2(log τ)xx.
KP
KP
(τ4x ? 4τxt + 3τyy)τ ? 4(τ3x ? τt )τx + 3(τxx ? τy)(τxx + τy) = 0

? ?
(N.C.Freeman and J.J.C.Nimmo, Phys.Lett.A 95(1983)1. )
τ(x, y, t) = det
?
??????????????????????
f
(0)
1
f
(1)
1
· · · f
(N?1)
1
f
(0)
2
f
(1)
2
· · · f
(N?1)
2
...
...
...
...
f
(0)
N
f
(1)
N
· · · f
(N?1)
N
?
??????????????????????
, where f
(k)
i
:=
?k
fi
?xk
fi(x, y, t)
?fi
?y
=
?2
fi
?x2
,
?fi
?t
=
?3
fi
?x3
? ?

(1/2)
N = 2
τ(x, y, t) =
f1 f1
′
f2 f2
′ = |0 1|.
τx(x, y, t) =
?
?x
f1 f1
′
f2 f2
′ =
f′
1
f′
1
f′
2
f′
2
+
f1 f1
′′
f2 f2
′′ = |0 2|
?f
?y
= ?2
f
?x2 , ?f
?t
= ?3
f
?x3
τy(x, y, t) =
?
?y
f1 f′
1
f2 f′
2
=
f′′
1
f′
1
f′′
2
f′
2
+
f1 f′′′
1
f2 f′′′
2
= |2 1| + |0 3|
τt (x, y, t) =
?
?t
f1 f′
1
f2 f′
2
=
f′′′
1
f′
1
f′′′
2
f′
2
+
f1 f′′′′
1
f2 f′′′′
2
= |3 1| + |0 4|

2/2
KP
(τ4x ? 4τxt + 3τyy )τ ? 4(τ3x ? τt )τx + 3(τxx ? τy )(τxx + τy )
=12(|0 1| × |2 3| ? |0 2| × |1 3| + |0 3| × |1 2|)
=12
f1 f′
1
f2 f′
2
f′′
1
f′′′
1
f′′
2
f′′′
2
?
f1 f′′
1
f2 f′′
2
f′
1
f′′′
1
f′
2
f′′′
2
+
f1 f′′′
1
f2 f′′′
2
f′
1
f′′
1
f′
2
f′′
2
Pl¨ucker 0
τ(x, y, t) KP
? ?
(One of ) the Pl¨ucker relations is expressed by
|a1 a2 · · · aN?2 b1 b2||a1 a2 · · · aN?2 b3 b4|
?|a1 a2 · · · aN?2 b1 b3||a1 a2 · · · aN?2 b2 b4|
+|a1 a2 · · · aN?2 b1 b4||a1 a2 · · · aN?2 b2 b3| = 0,
where ai , bi are arbitrary Nth column vectors.
? ?

KP
ex1) N = 2
fi(x, y, t) = exp(pix + p2
i
y + p3
i
t) + exp(qix + q2
i
y + q3
i
t)
1
ex2) N = 3, M = 6
fi(x, y, t) =
M
j=1
cijeθj
, θj = pjx + p2
j
y + p3
j
t
2 (Y. Kodama, J. Phys. A:Math. Theor. 43 (2010)434004)

1-2
(R. Hirota, “Nonlinear Partial Difference Equations. I, II, III, IV, V”, JPSJ (1977)).
KdV (in bilinear form)
3f2
xx ? fxft ? 4fxf3x + fftx + ff4x = 0 (f = f(x, t))
KdV (in bilinear form)
fm+1
n+1
fm?1
n
= (1 ? δ)fm
n+1
fm
n + δfm?1
n+1
fm+1
n
(fm
n = f(m, n))
m, n δ

KdV
KdV
fm+1
n+1
fm?1
n
= (1 ? δ)fm
n+1
fm
n + δfm?1
n+1
fm+1
n
2-
fm
n = 1 + eη1
+ eη2
+ a12eη1+η2
,
ηi = pim ? qin + ci
qi = log
δ + epi
1 + δepi
, a12 =
ep1
? ep2
?1 + ep1+p2
2
pi, ci
KdV 2-
um
n = fm
n+1
fm+1
n
/fm
n /fm+1
n+1

KP
KP
KP
a1(a2 ? a3)τ(l + 1, m, n)τ(l, m + 1, n + 1)
+a2(a3 ? a1)τ(l, m + 1, n)τ(l + 1, m, n + 1)
+a3(a1 ? a2)τ(l, m, n + 1)τ(l + 1, m + 1, n) = 0
a1, a2, a3

KP
KP KP (
)
τ(l, m, n) = det
?
?????????????????
?1(0) ?1(1) · · · ?1(N ? 1)
?2(0) ?2(1) · · · ?2(N ? 1)
...
...
...
...
?N(0) ?N(1) · · · ?N(N ? 1)
?
?????????????????
?i(s) = ?i(l, m, n, s) s ?i(s)
?i(l + 1, m, n, s) = ?i(l, m, n, s) + a1?i(l, m, n, s + 1)
?i(l, m + 1, n, s) = ?i(l, m, n, s) + a2?i(l, m, n, s + 1)
?i(l, m, n + 1, s) = ?i(l, m, n, s) + a3?i(l, m, n, s + 1)

1-3
1990 .
(T. Tokihiro et al. Phys. Rev. Lett. 76 (1996))
40 20 0 20 40
n
1.2
1.4
1.6
1.8
2.0
u
ultradiscretization
????????????????→
10 5 5 10
n
0.5
0.5
1.0
1.5
2.0
U

2
xn+1 =
a + xn
xn?1
(x0, x1, a > 0)
(
(1)
xn = eXn/?
, a = eA/?
(2) lim?→+0 ? log
Xn+1 = lim
?→+0
? log eA/?
+ eXn/?
? Xn?1
lim
?→+0
? log eA/?
+ eB/?
= max(A, B)
Xn+1 = max(A, Xn) ? Xn?1.

xn+1 =
a + xn
xn?1
→ Xn+1 = max(A, Xn) ? Xn?1.
+ → max
? → not well-de?ned
× → +
÷ → ?

lim
?→+0
? log eA/?
+ eB/?
= max(A, B)
lim
?→+0
? log eA/?
?eB/?
=
?
???
???
A (A > B)
(A ≤ B)

KdV
KdV
fm+1
n+1
fm?1
n
= (1 ? δ)fm
n+1
fm
n + δfm?1
n+1
fm+1
n
fm
n = eFm
n
/?
, δ = e?2/ε
KdV (bilinear form)
ultradiscretization
?????????→ Fm+1
n+1
+ Fm?1
n
= max(Fm
n+1
+ Fm
n , Fm?1
n+1
+ Fm+1
n
? 2)

KdV
2-
fm
n = 1 + eη1
+ eη2
+ a12eη1+η2
ηi(m, n) = pim ? qin + ci
qi = log
δ + epi
1 + δepi
, a12 =
ep1
? ep2
?1 + ep1+p2
2
pi = ePi /?
, qi = eQi /?
, ci = eCi /?
, δ = e?2/?
2-
Fm
n = max(0, S1, S2, S1 + S2 ? A12),
Si(m, n) = Pim ? Qin + Ci
Qi =
1
2
(|Pi + 1| ? |Pi ? 1|), A12 = |P1 + P2| ? |P1 ? P2|
KdV

KdV 2-
Fm
n = max(0, 3m ? n, m ? n + 1, 4m ? 2n ? 1),
Um
n =Fm
n+1
+ Fm+1
n
? Fm
n ? Fm+1
n+1

→
???????→
???????→
???????→
Pl¨ucker
2
det
a b
c d
= ad?bc.
3

UP
2.1.
? ?
(UP)
N A = [aij]1≤i,j≤N A
(UP) . (D. Takahashi, R. Hirota, “Ultradiscrete Soliton Solution of
Permanent Type”, J. Phys. Soc. Japan, 76 (2007) 104007–104012)
up[A] ≡ max
π∈SN
1≤i≤N
aiπi
maxπ∈SN
N π = (π1, π2, . . . , πN)
.
? ?
cf)
det[A] ≡
π∈SN 1≤i≤N
sgn(π)aiπi
perm[A] ≡
π∈SN 1≤i≤N
aiπi

UP
UP
UP
2 × 2 matrix
up
a11 a12
a21 a22
= max (a11 + a22, a12 + a21)
3 × 3 matrix
up
?
?????????
a11 a12 a13
a21 a22 a23
a31 a32 a33
?
?????????
= max a11 + a22 + a33, a11 + a23 + a32, a12 + a21 + a33,
a12 + a23 + a31, a13 + a21 + a32, a13 + a22 + a31

UP
UP
c × det
?
?????????
a11 a12 a13
a21 a22 a33
a31 a32 a33
?
?????????
= det
?
?????????
ca11 a12 a13
ca21 a22 a33
ca31 a32 a33
?
?????????
(c : const.)
det
?
?????????
a11 + b1 a12 a13
a21 + b2 a22 a33
a31 + b3 a32 a33
?
?????????
= det
?
?????????
a11 a12 a13
a21 a22 a33
a31 a32 a33
?
?????????
+ det
?
?????????
b1 a12 a13
b2 a22 a33
b3 a32 a33
?
?????????

UP
c + up
?
?????????
a11 a12 a13
a21 a22 a33
a31 a32 a33
?
?????????
= up
?
?????????
c + a11 a12 a13
c + a21 a22 a33
c + a31 a32 a33
?
?????????
up
?
?????????
max(a11, b1) a12 a13
max(a21, b2) a22 a33
max(a31, b3) a32 a33
?
?????????
= max
?
??????up
?
?????????
a11 a12 a13
a21 a22 a33
a31 a32 a33
?
?????????
, up
?
?????????
b1 a12 a13
b2 a22 a33
b3 a32 a33
?
?????????
?
??????

UP
UP det
det
a11 + a12 a12 + a13
a21 + a22 a22 + a23
= det
a11 a12
a21 a22
+ det
a11 a13
a21 a23
+ det
a12 a13
a22 a23

up
max(a11, a12) max(a12, a13)
max(a21, a22) max(a22, a23)
= max
?
??????up
a11 a12
a21 a22
, up
a11 a13
a21 a23
, up
a12 a12
a22 a22
, up
a12 a13
a22 a23
?
??????
UP

UP
UP
KdV
(D. Takahashi, R. Hirota, “Ultradiscrete Soliton Solution of Permanent Type”, J. Phys. Soc. Japan,
76 (2007) 104007–104012)
(H. Nagai, “ A new expression of a soliton solution to the ultradiscrete Toda equation”, J. Phys. A:
Math. Theor. 41 (2008) 235204(12pp))
KP
(H. Nagai and D. Takahashi, “Ultradiscrete Pl¨ucker Relation Specialized for Soliton Solutions”, J.
Phys. A: Math. Theor. 44 (2011) 095202(18pp))
hungry-Lotka Volterra
(S. Nakamura, “Ultradiscrete soliton equations derived from ultradiscrete permanent formulae”, J.
Phys. A: Math. Theor. 44 (2011) 295201(14pp))

UP
KP UP
KP
T(l, m + 1, n) + T(l + 1, m, n + 1)
= max(T(l + 1, m, n) + T(l, m + 1, n + 1) ? A1 + A2,
T(l, m, n + 1) + T(l + 1, m + 1, n)) (A1 ≥ A2)
UP (H.Nagai, arXiv:nlin:1611.09081)
T(l, m, n) = up
?
??????????????????
?1(0) ?1(1) · · · ?1(N ? 1)
?2(0) ?2(1) · · · ?2(N ? 1)
...
...
...
...
?N(0) ?N(1) · · · ?N(N ? 1)
?
??????????????????
?i(s) = ?i(l, m, n, s) s l, m, n
3

UP
UP ?i(s)
1 A1 ≥ A2 ≥ A3
?i(l + 1, m, n; s) = max(?i(l, m, n; s), ?i(l, m, n; s + 1) ? A1)
?i(l, m + 1, n; s) = max(?i(l, m, n; s), ?i(l, m, n; s + 1) ? A2)
?i(l, m, n + 1; s) = max(?i(l, m, n; s), ?i(l, m, n; s + 1) ? A3)
2 j, i1, i2
?i1
(s + j) + ?i2
(s + j)
≤ max ?i1
(s + j ? 1) + ?i2
(s + j + 1), ?i2
(s + j ? 1) + ?i1
(s + j + 1)
3 (?1(s), ?2(s), . . . , ?N(s))T
= Φ(s) ,
0 ≤ k1 < k2 < k3 ≤ N + 1
up[Φ(0) · · · Φ(k2) · · · Φ(N)] + up[Φ(0) · · · Φ(k1) · · · Φ(k3) · · · Φ(N + 1)]
= max up[Φ(0) · · · Φ(k3) · · · Φ(N)] + up[Φ(0) · · · Φ(k1) · · · Φ(k2) · · · Φ(N + 1)]

UP
KP
τ(x, y, t) = det
?
?????????????????
f1 f′
1
· · · f
(N?1)
1
.
.
.
.
.
.
...
.
.
.
fN f′
N
· · · f
(N?1)
N
?
?????????????????
?
???????
fj = fj (x, y, t)
?fj
?y
=
?2fj
?x2
,
?fj
?t
=
?3fj
?x3
.
?
???????
KP
1
12
(ττ4x ? 4τx τ3x + 3τ2
xx ) ?
1
3
(ττxt ? τx τt ) +
1
4
(ττyy ? τ2
y ) = 0
Pl¨ucker n = 3
|a1 . . . aN?2 b1 b2| × |a1 . . . aN?2 b3 b4|
?|a1 . . . aN?2 b1 b3| × |a1 . . . aN?2 b2 b4|
+|a1 . . . aN?2 b1 b4| × |a1 . . . aN?2 b2 b3| = 0

UP
KP
τ(l, m, n) = det
?
??????????????
φ1(0) φ1(1) · · · φ1(N ? 1)
.
.
.
.
.
.
...
.
.
.
φN (0) φN (1) · · · φN (N ? 1)
?
??????????????
?
????????
φj (s) = φj (s; l, m, n)
.
?
????????
KP
a1(a2 ? a3)τ(l + 1, m, n)τ(l, m + 1, n + 1)
+a2(a3 ? a1)τ(l, m + 1, n)τ(l + 1, m, n + 1)
+a3(a1 ? a2)τ(l, m, n + 1)τ(l + 1, m + 1, n) = 0
Pl¨ucker n = 3
|a1 . . . aN?2 b1 b2| × |a1 . . . aN?2 b3 b4|
?|a1 . . . aN?2 b1 b3| × |a1 . . . aN?2 b2 b4|
+|a1 . . . aN?2 b1 b4| × |a1 . . . aN?2 b2 b3| = 0

UP
KP
KP UP
T(l, m, n) = up
?
??????????????
φ1(0) φ1(1) · · · φ1(N ? 1)
.
.
.
.
.
.
...
.
.
.
φN (0) φN (1) · · · φN (N ? 1)
?
??????????????
KP
T(l, m + 1, n) + T(l + 1, m, n + 1)
= max T(l + 1, m, n) + T(l, m + 1, n + 1) ? A1 + A2,
T(l, m, n + 1) + T(l + 1, m + 1, n) (A1 > A2)
3
= max up[Φ(0) · · · Φ(k3) · · · Φ(N)] + up[Φ(0) · · · Φ(k1) · · · Φ(k2) · · · Φ(N + 1)]

UP
1, 2
?i(s) 1 ≤ i ≤ N
?i(l + 1, m, n; s) = max(?i(l, m, n; s), ?i(l, m, n; s + 1) ? A1)
?i(l, m + 1, n; s) = max(?i(l, m, n; s), ?i(l, m, n; s + 1) ? A2)
?i(l, m, n + 1; s) = max(?i(l, m, n; s), ?i(l, m, n; s + 1) ? A3)
T(l + 1, m, n) 2N
ex) N = 2
T(l + 1, m, n) = up
?1(l + 1; 0) ?1(l + 1; 1)
?2(l + 1; 0) ?2(l + 1; 1)
= max up
?1(0) ?1(1)
?2(0) ?2(1)
, up
?1(1) ?1(1)
?2(1) ?2(1)
? A1,
up
?1(0) ?1(2)
?2(0) ?2(2)
? A1, up
?1(1) ?1(2)
?2(1) ?2(2)
? 2A1

UP
2
?i(s) 1 ≤ i1, i2 ≤ N 2
?i1
(s + j) + ?i2
(s + j)
≤ max ?i1
(s + j ? 1) + ?i2
(s + j + 1), ?i2
(s + j ? 1) + ?i1
(s + j + 1)
up
?1(s + 1) ?1(s + 1)
?2(s + 1) ?2(s + 1)
≤ up
?1(s) ?1(s + 2)
?2(s) ?2(s + 2)
UP
ex) N = 2
T(l + 1, m, n, s)
= max up
?1(0) ?1(1)
?2(0) ?2(1)
, up
?1(0) ?1(2)
?2(0) ?2(2)
? A1, up
?1(1) ?1(2)
?2(1) ?2(2)
? 2A1

UP
1
? ?
Theorem
The UP solution to the uKP equation is given by
T(l, m, n) = up
?
?????????????????
?1(0) ?1(1) · · · ?1(N ? 1)
?2(0) ?2(1) · · · ?2(N ? 1)
...
...
...
...
?N(0) ?N(1) · · · ?N(N ? 1)
?
?????????????????
?i (l, m, n, s)
= max Pi s + max(0, Pi ? A1)l + max(0, Pi ? A2)m + max(0, Pi ? A3)n + Ci ,
? Pi s + max(0, ?Pi ? A1)l + max(0, ?Pi ? A2)m + max(0, ?Pi ? A3)n + C′
i
where Pi, Ci and C′
i
are arbitrary parameters.(H.Nagai and D.Takahashi,
J.Phys.A Math. Theor. 44(2011))
? ?

UP
2
? ?
Theorem
The UP solution to the uKP equation is given by
T(l, m, n) = up
?
?????????????????
?1(0) ?1(1) · · · ?1(N ? 1)
?2(0) ?2(1) · · · ?2(N ? 1)
...
...
...
...
?N(0) ?N(1) · · · ?N(N ? 1)
?
?????????????????
?i (l, m, n, s) = max Ci1 + P1s + max(0, P1 ? A1)l + max(0, P1 ? A2)m + max(0, P1 ? A3)n,
Ci2 + P2s + max(0, P2 ? A1)l + max(0, P2 ? A2)m + max(0, P2 ? A3)n,
Ci3 + P3s + max(0, P3 ? A1)l + max(0, P3 ? A2)m + max(0, P3 ? A3)n
where Cij and Pj are arbitrary parameters. (H.Nagai, arXiv:nlin:1611.09081)
? ?

UP
31-32
2012,
(http://gcoe-
mi.jp/english/temp/publish/132702cf8b5f107c34bcc3c5077464ff.pdf)
B. Grammaticos, Y. Kosmann-Schwarzbach, and T. Tamizhmani
(Eds.), “Discrete Integrable Systems”, Lecture Notes in Physics,
Springer
Peter Butkovi?c, “Max-linear Systems: Theory and Algorithms”,
Springer

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