46
എ ܠύʔϚωϯτ ιϦτϯఔͱύʔϚωϯτ Ҫ ल༑ ౦ւେ ཧਓηϛφʔ@Ҵେ 2016 12 15 Ҫ ल༑ ιϦτϯఔͱύʔϚωϯτ
@2016 12 15
1.1.11.21.3
2.2.12.22.3
?: soliton
!
!
21965 N. Zabusky M. Kruskal
KdV (KdV: Korteweg-de Vries) 2-onsolitary wave(-on)
: solitron
Wikipedia
191950 60
1967
1970
1980
1990...
! KdV
ut
+ 6uux
+3ux3
= 0, (u = u(x, t))
! mKdV
ut
+ 6u2ux
+3ux3
= 0, (u = u(x, t))
! KP
x
(4ut 6uu
x
3ux3
) 3
2uy2
= 0, (u = u(x, y, t))
!
d2
dt2log(1 + Vn) = Vn+1 2Vn + Vn1 (Vn = Vn(t))
KdV KP .
KdV
KdV
ut
+ 6uux
+3ux3
= 0, (u = u(x, t)).
u(x, t) = 2k 2 sech2 k(x 4k 2t + c), k , c
file:///Users/naoya/Desktop/MyWork/%E6%95%B0%E7%90%86%E4%BA%BA%E3%82%BB%E3%83%9F%E3%83%8A%E3%83%BC/20161215_%E9%95%B7%E4%BA%95%E3%81%95%E3%82%93/%E3%82%B9%E3%83%A9%E3%82%A4%E3%83%88%E3%82%99/./movie/KdV1movie.AVI
KdV 2-
KdV2-
u(x, t) = 22
x2log f(x, t)
f(x, t) = 1+e1+e2+(
k1 k2k1 + k2
)2e1+2 , j(x, t) = kjxk 3j t+cj
2-
file:///Users/naoya/Desktop/MyWork/%E6%95%B0%E7%90%86%E4%BA%BA%E3%82%BB%E3%83%9F%E3%83%8A%E3%83%BC/20161215_%E9%95%B7%E4%BA%95%E3%81%95%E3%82%93/%E3%82%B9%E3%83%A9%E3%82%A4%E3%83%88%E3%82%99/./movie/KdV2movie.AVI
KP
Kadomtsev-Petviashvili (KP)
x
(4ut 6uu
x
3ux3
) 3
2uy2
= 0, (u = u(x, y, t))
2-
u(x, y, t) = 22
x2log (x, y, t)
(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 = p2i q2i, i = p3i q
3i
(i = 1, 2)
pi , qi KP
file:///Users/naoya/Desktop/MyWork/%E6%95%B0%E7%90%86%E4%BA%BA%E3%82%BB%E3%83%9F%E3%83%8A%E3%83%BC/20161215_%E9%95%B7%E4%BA%95%E3%81%95%E3%82%93/%E3%82%B9%E3%83%A9%E3%82%A4%E3%83%88%E3%82%99/./movie/KP2otype.AVI
KP
x
(4ut 6uu
x
3ux3
) 3
2uy2
= 0, (u = u(x, y, t))
u = 2(log )xx .
KPKP
(4x 4xt + 3yy) 4(3x t)x + 3(xx y)(xx + y) = 0
(N.C.Freeman and J.J.C.Nimmo, Phys.Lett.A 95(1983)1. )
(4x 4xt + 3yy) 4(3x t)x + 3(xx y)(xx + y) = 0
(x, y, t) = det
f (0)1
f (1)1
f (N1)1
f (0)2
f (1)2
f (N1)2
....... . .
...
f (0)N
f (1)N
f (N1)N
, where f (k)i
:=k fixk
fi(x, y, t)
fiy
=2fix2,
fit
=3fix3
(1/2)
N = 2
(x, y, t) =f1 f1f2 f2
= |0 1|.
x(x, y, t) =
x
f1 f1f2 f2
=f 1
f 1
f 2
f 2
+f1 f1f2 f2
= |0 2|
fy =
2fx2 ,
ft =
3fx3
y(x, y, t) =
y
f1 f 1f2 f 2
=f 1
f 1
f 2
f 2
+f1 f 1f2 f 2
= |2 1|+ |0 3|
t(x, y, t) =
t
f1 f 1f2 f 2
=f 1
f 1
f 2
f 2
+f1 f 1f2 f 2
= |3 1|+ |0 4|
2/2
KP(4x 4xt + 3yy) 4(3x t)x + 3(xx y)(xx + y)
=12(|0 1| |2 3| |0 2| |1 3|+ |0 3| |1 2|)
=12(
f1 f 1f2 f 2
f 1
f 1
f 2
f 2
f1 f 1f2 f 2
f 1
f 1
f 2
f 2
+f1 f 1f2 f 2
f 1
f 1
f 2
f 2
)
Plucker 0(x, y, t) KP
(One of ) the Plucker relations is expressed by
|a1 a2 aN2 b1 b2||a1 a2 aN2 b3 b4||a1 a2 aN2 b1 b3||a1 a2 aN2 b2 b4|+|a1 a2 aN2 b1 b4||a1 a2 aN2 b2 b3| = 0,
where ai , bi are arbitrary Nth column vectors.
KP
ex1) N = 2
fi(x, y, t) = exp(pix + p2i y + p3it) + exp(qix + q2i y + q
3it)
1
ex2) N = 3, M = 6
fi(x, y, t) =M
j=1
cijej , j = pjx + p2j y + p3jt
2 (Y. Kodama, J. Phys. A:Math. Theor. 43 (2010)434004)
file:///Users/naoya/Desktop/MyWork/%E6%95%B0%E7%90%86%E4%BA%BA%E3%82%BB%E3%83%9F%E3%83%8A%E3%83%BC/20161215_%E9%95%B7%E4%BA%95%E3%81%95%E3%82%93/%E3%82%B9%E3%83%A9%E3%82%A4%E3%83%88%E3%82%99/./movie/KP2otype.AVIfile:///Users/naoya/Desktop/MyWork/%E6%95%B0%E7%90%86%E4%BA%BA%E3%82%BB%E3%83%9F%E3%83%8A%E3%83%BC/20161215_%E9%95%B7%E4%BA%95%E3%81%95%E3%82%93/%E3%82%B9%E3%83%A9%E3%82%A4%E3%83%88%E3%82%99/./movie/KP3.AVI
1-2
(R. Hirota, Nonlinear Partial Difference Equations. I, II, III, IV, V, JPSJ (1977)).KdV (in bilinear form)
3f2xx fx ft 4fx f3x + fftx + ff4x = 0 (f = f(x, t))
KdV (in bilinear form)
fm+1n+1
fm1n = (1 )fmn+1fmn + f
m1n+1
fm+1n (fmn = f(m, n))
m, n
KdV
KdV
fm+1n+1
fm1n = (1 )fmn+1fmn + f
m1n+1
fm+1n
2-fmn = 1 + e
1 + e2 + a12e1+2 ,
i = pim qin + ci
qi = log( + epi
1 + epi
), a12 =
( ep1 ep21 + ep1+p2
)2
pi , ciKdV 2-
umn = fmn+1f
m+1n /f
mn /f
m+1n+1
file:///Users/naoya/Desktop/MyWork/%E6%95%B0%E7%90%86%E4%BA%BA%E3%82%BB%E3%83%9F%E3%83%8A%E3%83%BC/20161215_%E9%95%B7%E4%BA%95%E3%81%95%E3%82%93/%E3%82%B9%E3%83%A9%E3%82%A4%E3%83%88%E3%82%99/./movie/dKdV1movie.AVI
KP
KP
(4x 4xt + 3yy) 4(3x t)x + 3(xx y)(xx + y) = 0
KP
a1(a2 a3)(l + 1,m, n)(l,m + 1, n + 1)+a2(a3 a1)(l,m + 1, n)(l + 1,m, n + 1)+a3(a1 a2)(l,m, n + 1)(l + 1,m + 1, n) = 0
a1, a2, a3
KP
KP KP ()
(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) + a1i(l,m, n, s + 1)i(l,m + 1, n, s) = i(l,m, n, s) + a2i(l,m, n, s + 1)i(l,m, n + 1, s) = i(l,m, n, s) + a3i(l,m, n, s + 1)
1-3
1990 .
(T. Tokihiro et al. Phys. Rev. Lett. 76 (1996))
!!!!!!!!!!!!!!!!!
!
!
!
!
!
!
!!!
!
!
!
!
!
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!40 !20 0 20 40 n1.2
1.4
1.6
1.8
2.0u
ultradiscretization
! !
! ! !
! ! ! !
!
! ! ! ! ! ! ! ! ! ! !
!10 !5 5 10 n!0.5
0.51.01.52.0U
2
xn+1 =a + xn
xn1(x0, x1, a > 0)
(
(1)xn = eXn/ , a = eA/
(2) lim+0 log
Xn+1 = lim+0
log(eA/ + eXn/
) Xn1
lim+0
log(eA/ + eB/
)= max(A ,B)
Xn+1 = max(A , Xn) Xn1.
xn+1 =a + xn
xn1 Xn+1 = max(A , Xn) Xn1.
+ max not well-defined +
xn+1 =a + xn
xn1 Xn+1 = max(A , Xn) Xn1.
+ max not well-defined +
!
!
lim+0
log(eA/ + eB/
)= max(A ,B)
lim+0
log(eA/eB/
)=
A (A > B)(A B)
KdV
KdV
fm+1n+1
fm1n = (1 )fmn+1fmn + f
m1n+1
fm+1n
fmn = eFmn / , = e2/
KdV (bilinear form)
ultradiscretization Fm+1n+1
+ Fm1n = max(Fmn+1 + F
mn , F
m1n+1
+ Fm+1n 2)
KdV
2-fmn = 1 + e
1 + e2 + a12e1+2
i(m, n) = pim qin + ci
qi = log( + epi
1 + epi
), a12 =
( ep1 ep21 + ep1+p2
)2
pi = ePi/ , qi = eQi/ , ci = eCi/ , = e2/
2-
Fmn = max(0, S1, S2, S1 + S2 A12),
Si(m, n) = Pim Qin + Ci
Qi =12(|Pi + 1| |Pi 1|), A12 = |P1 + P2| |P1 P2|
KdV
KdV 2-
Fmn = max(0, 3m n,m n + 1, 4m 2n 1),Umn =F
mn+1 + F
m+1n Fmn F
m+1n+1
file:///Users/naoya/Desktop/MyWork/%E6%95%B0%E7%90%86%E4%BA%BA%E3%82%BB%E3%83%9F%E3%83%8A%E3%83%BC/20161215_%E9%95%B7%E4%BA%95%E3%81%95%E3%82%93/%E3%82%B9%E3%83%A9%E3%82%A4%E3%83%88%E3%82%99/./movie/uKdV1movie.AVI
!
!
!
!
! max!
!
!
! Plucker
2
det[a bc d
]= adbc.
3
!
!
!
!
!
!
!
!
!
!
!
!
UP
2.1. (UP)
N A = [aij]1i,jN A(UP) . (D. Takahashi, R. Hirota, Ultradiscrete Soliton Solution ofPermanent Type, J. Phys. Soc. Japan, 76 (2007) 104007104012)
up[A] maxSN
1iNaii
maxSN N = (1, 2, . . . , N).
cf)det[A]
SN
1iNsgn()aii
perm[A]
SN
1iNaii
UP
UP
! UP2 2 matrix
up[a11 a12a21 a22
]= max (a11 + a22, a12 + a21)
3 3 matrix
up
a11 a12 a13a21 a22 a23a31 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 a13a21 a22 a33a31 a32 a33
= det
ca11 a12 a13ca21 a22 a33ca31 a32 a33
(c : const.)
det
a11 + b1 a12 a13a21 + b2 a22 a33a31 + b3 a32 a33
= det
a11 a12 a13a21 a22 a33a31 a32 a33
+ det
b1 a12 a13b2 a22 a33b3 a32 a33
***************************************************************************
UP
c + up
a11 a12 a13a21 a22 a33a31 a32 a33
= up
c + a11 a12 a13c + a21 a22 a33c + a31 a32 a33
up
max(a11, b1) a12 a13max(a21, b2) a22 a33max(a31, b3) a32 a33
= max
up
a11 a12 a13a21 a22 a33a31 a32 a33
, up
b1 a12 a13b2 a22 a33b3 a32 a33
UP
UP det
det[a11 + a12 a12 + a13a21 + a22 a22 + a23
]
= det[a11 a12a21 a22
]+ det
[a11 a13a21 a23
]+ det
[a12 a13a22 a23
]
***************************************************************************
up[max(a11, a12) max(a12, a13)max(a21, a22) max(a22, a23)
]
= maxup
[a11 a12a21 a22
], up
[a11 a13a21 a23
], up
[a12 a12a22 a22
], up
[a12 a13a22 a23
]
UP
UP
UP
! KdV(D. Takahashi, R. Hirota, Ultradiscrete Soliton Solution of Permanent Type, J. Phys. Soc. Japan,76 (2007) 104007104012)
!
(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 Plucker 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, n3
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 = (s) ,0 k1 < k2 < k3 N + 1
up[(0) (k2) (N)] + up[(0) (k1) (k3) (N + 1)]
= max(
up[(0) (k3) (N)] + up[(0) (k1) (k2) (N + 1)]
up[(0) (k1) (N)] + up[(0) (k2) (k3) (N + 1)])
UP
KP
(x, y, t) = det
f1 f 1 f(N1)1
.
.
....
. . ....
fN f N f(N1)N
fj = fj(x, y, t)fjy =
2 fjx2,
fjt =
3 fjx3.
KP
112
(4x 4x3x + 32xx) 13(xt xt ) +
14(yy 2y) = 0
Plucker n = 3
|a1 . . . aN2 b1 b2| |a1 . . . aN2 b3 b4||a1 . . . aN2 b1 b3| |a1 . . . aN2 b2 b4|+|a1 . . . aN2 b1 b4| |a1 . . . aN2 b2 b3| = 0
UP
KP
(l,m, n) = det
1(0) 1(1) 1(N 1)...
.
.
.. . .
.
.
.N(0) N(1) N(N 1)
j(s) = j(s; l,m, n)
.
KP
a1(a2 a3)(l + 1,m, n)(l,m + 1, n + 1)+a2(a3 a1)(l,m + 1, n)(l + 1,m, n + 1)+a3(a1 a2)(l,m, n + 1)(l + 1,m + 1, n) = 0
Plucker n = 3
|a1 . . . aN2 b1 b2| |a1 . . . aN2 b3 b4||a1 . . . aN2 b1 b3| |a1 . . . aN2 b2 b4|+|a1 . . . aN2 b1 b4| |a1 . . . aN2 b2 b3| = 0
UP
KPKP UP
T(l,m, n) = up
1(0) 1(1) 1(N 1)...
.
.
.. . .
.
.
.N(0) N(1) N(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
up[(0) (k2) (N)] + up[(0) (k1) (k3) (N + 1)]
= max(
up[(0) (k3) (N)] + up[(0) (k1) (k2) (N + 1)]
up[(0) (k1) (N)] + up[(0) (k2) (k3) (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 2i1(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
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 = (s) ,0 k1 < k2 < k3 N + 1
up[(0) (k2) (N)] + up[(0) (k1) (k3) (N + 1)]
= max(
up[(0) (k3) (N)] + up[(0) (k1) (k2) (N + 1)]
up[(0) (k1) (N)] + up[(0) (k2) (k3) (N + 1)])
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(Pis + max(0, Pi A1)l + max(0, Pi A2)m + max(0, Pi A3)n + Ci ,
Pis + 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
KP KPUP
!
! max-plus
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 Butkovic, Max-linear Systems: Theory and Algorithms,Springer
wigUgU
Up[}lg`UPUp[}lg
