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Hidetomo Nagai

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Hidetomo Nagai

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Hidetomo Nagai

  1. 1. @ 2016 12 15
  2. 2. 1. 1.1 1.2 1.3 2. 2.1 2.2 2.3
  3. 3. ? : soliton 2 1965 N. Zabusky M. Kruskal KdV (KdV: Korteweg-de Vries) 2 -on solitary wave(-on) : solitron Wikipedia
  4. 4. 19 1950 60 → 1967 1970 1980 → 1990 ...
  5. 5. 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 .
  6. 6. 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
  7. 7. 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-
  8. 8. 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+Ω1t + eP2x+Q2y+Ω2t + (p1 − p2)(q1 − q2) (p1 − q2)(q1 − p2) e(P1+P2)x+(Q1+Q2)y+(Ω+Ω2)t , Pi = pi − qi, Qi = p2 i − q2 i , Ωi = p3 i − q3 i (i = 1, 2) pi, qi KP
  9. 9. 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
  10. 10. ✓ ✏ (N.C.Freeman and J.J.C.Nimmo, Phys.Lett.A 95(1983)1. ) (τ4x − 4τxt + 3τyy)τ − 4(τ3x − τt )τx + 3(τxx − τy)(τxx + τy) = 0 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 ✒ ✑
  11. 11. (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|
  12. 12. 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. ✒ ✑
  13. 13. 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)
  14. 14. 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
  15. 15. 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
  16. 16. KP KP (τ4x − 4τxt + 3τyy)τ − 4(τ3x − τt )τx + 3(τxx − τy)(τxx + τy) = 0 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
  17. 17. KP KP KP ( ) τ(l, m, n) = det ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ϕ1(0) ϕ1(1) · · · ϕ1(N − 1) ϕ2(0) ϕ2(1) · · · ϕ2(N − 1) ... ... ... ... ϕN(0) ϕN(1) · · · ϕN(N − 1) ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ϕi(s) = ϕi(l, m, n, s) s ϕi(s) ϕi(l + 1, m, n, s) = ϕi(l, m, n, s) + a1ϕi(l, m, n, s + 1) ϕi(l, m + 1, n, s) = ϕi(l, m, n, s) + a2ϕi(l, m, n, s + 1) ϕi(l, m, n + 1, s) = ϕi(l, m, n, s) + a3ϕi(l, m, n, s + 1)
  18. 18. 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
  19. 19. 2 xn+1 = a + xn xn−1 (x0, x1, a > 0) ( (1) xn = eXn/ϵ , a = eA/ϵ (2) limϵ→+0 ϵ log Xn+1 = lim ϵ→+0 ϵ log eA/ϵ + eXn/ϵ − Xn−1 lim ϵ→+0 ϵ log eA/ϵ + eB/ϵ = max(A, B) Xn+1 = max(A, Xn) − Xn−1.
  20. 20. xn+1 = a + xn xn−1 → Xn+1 = max(A, Xn) − Xn−1. + → max − → not well-defined × → + ÷ → −
  21. 21. xn+1 = a + xn xn−1 → Xn+1 = max(A, Xn) − Xn−1. + → max − → not well-defined × → + ÷ → −
  22. 22. lim ϵ→+0 ϵ log eA/ϵ + eB/ϵ = max(A, B) lim ϵ→+0 ϵ log eA/ϵ −eB/ϵ = ⎧ ⎪⎪⎨ ⎪⎪⎩ A (A > B) (A ≤ B)
  23. 23. 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 /ϵ , 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)
  24. 24. 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 /ϵ , qi = eQi /ϵ , ci = eCi /ϵ , δ = e−2/ϵ 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
  25. 25. 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
  26. 26. max
  27. 27. → −−−−−−−→ −−−−−−−→ −−−−−−−→ Pl¨ucker 2 det a b c d = ad−bc. 3
  28. 28. 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
  29. 29. 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
  30. 30. 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 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ ⎞ ⎟⎟⎟⎟⎟⎠
  31. 31. 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
  32. 32. 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))
  33. 33. 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 ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ϕ1(0) ϕ1(1) · · · ϕ1(N − 1) ϕ2(0) ϕ2(1) · · · ϕ2(N − 1) ... ... ... ... ϕN(0) ϕN(1) · · · ϕN(N − 1) ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ϕi(s) = ϕi(l, m, n, s) s l, m, n 3
  34. 34. UP UP ϕi(s) 1 A1 ≥ A2 ≥ A3 ϕi(l + 1, m, n; s) = max(ϕi(l, m, n; s), ϕi(l, m, n; s + 1) − A1) ϕi(l, m + 1, n; s) = max(ϕi(l, m, n; s), ϕi(l, m, n; s + 1) − A2) ϕi(l, m, n + 1; s) = max(ϕi(l, m, n; s), ϕi(l, m, n; s + 1) − A3) 2 j, i1, i2 ϕi1 (s + j) + ϕi2 (s + j) ≤ max ϕi1 (s + j − 1) + ϕi2 (s + j + 1), ϕi2 (s + j − 1) + ϕi1 (s + j + 1) 3 (ϕ1(s), ϕ2(s), . . . , ϕN(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[Φ(0) · · · Φ(k1) · · · Φ(N)] + up[Φ(0) · · · Φ(k2) · · · Φ(k3) · · · Φ(N + 1)]
  35. 35. 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
  36. 36. 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
  37. 37. 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 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[Φ(0) · · · Φ(k1) · · · Φ(N)] + up[Φ(0) · · · Φ(k2) · · · Φ(k3) · · · Φ(N + 1)]
  38. 38. UP 1, 2 ϕi(s) 1 ≤ i ≤ N ϕi(l + 1, m, n; s) = max(ϕi(l, m, n; s), ϕi(l, m, n; s + 1) − A1) ϕi(l, m + 1, n; s) = max(ϕi(l, m, n; s), ϕi(l, m, n; s + 1) − A2) ϕi(l, m, n + 1; s) = max(ϕi(l, m, n; s), ϕi(l, m, n; s + 1) − A3) T(l + 1, m, n) 2N ex) N = 2 T(l + 1, m, n) = up ϕ1(l + 1; 0) ϕ1(l + 1; 1) ϕ2(l + 1; 0) ϕ2(l + 1; 1) = max up ϕ1(0) ϕ1(1) ϕ2(0) ϕ2(1) , up ϕ1(1) ϕ1(1) ϕ2(1) ϕ2(1) − A1, up ϕ1(0) ϕ1(2) ϕ2(0) ϕ2(2) − A1, up ϕ1(1) ϕ1(2) ϕ2(1) ϕ2(2) − 2A1
  39. 39. UP 2 ϕi(s) 1 ≤ i1, i2 ≤ N 2 ϕi1 (s + j) + ϕi2 (s + j) ≤ max ϕi1 (s + j − 1) + ϕi2 (s + j + 1), ϕi2 (s + j − 1) + ϕi1 (s + j + 1) up ϕ1(s + 1) ϕ1(s + 1) ϕ2(s + 1) ϕ2(s + 1) ≤ up ϕ1(s) ϕ1(s + 2) ϕ2(s) ϕ2(s + 2) UP ex) N = 2 T(l + 1, m, n, s) = max up ϕ1(0) ϕ1(1) ϕ2(0) ϕ2(1) , up ϕ1(0) ϕ1(2) ϕ2(0) ϕ2(2) − A1, up ϕ1(1) ϕ1(2) ϕ2(1) ϕ2(2) − 2A1
  40. 40. UP UP ϕi(s) 1 A1 ≥ A2 ≥ A3 ϕi(l + 1, m, n; s) = max(ϕi(l, m, n; s), ϕi(l, m, n; s + 1) − A1) ϕi(l, m + 1, n; s) = max(ϕi(l, m, n; s), ϕi(l, m, n; s + 1) − A2) ϕi(l, m, n + 1; s) = max(ϕi(l, m, n; s), ϕi(l, m, n; s + 1) − A3) 2 j, i1, i2 ϕi1 (s + j) + ϕi2 (s + j) ≤ max ϕi1 (s + j − 1) + ϕi2 (s + j + 1), ϕi2 (s + j − 1) + ϕi1 (s + j + 1) 3 (ϕ1(s), ϕ2(s), . . . , ϕN(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[Φ(0) · · · Φ(k1) · · · Φ(N)] + up[Φ(0) · · · Φ(k2) · · · Φ(k3) · · · Φ(N + 1)]
  41. 41. UP 1 ✓ ✏ Theorem The UP solution to the uKP equation is given by T(l, m, n) = up ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ϕ1(0) ϕ1(1) · · · ϕ1(N − 1) ϕ2(0) ϕ2(1) · · · ϕ2(N − 1) ... ... ... ... ϕN(0) ϕN(1) · · · ϕN(N − 1) ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ϕi (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)) ✒ ✑
  42. 42. UP 2 ✓ ✏ Theorem The UP solution to the uKP equation is given by T(l, m, n) = up ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ϕ1(0) ϕ1(1) · · · ϕ1(N − 1) ϕ2(0) ϕ2(1) · · · ϕ2(N − 1) ... ... ... ... ϕN(0) ϕN(1) · · · ϕN(N − 1) ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ϕi (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) ✒ ✑
  43. 43. UP KP KP UP max-plus
  44. 44. 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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