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No.16ME-S1 Reports of RIAM Symposium No.16ME-S1 Physics and Mathematical Structures of Nonlinear Waves Proceedings of a symposium held at Chikushi Campus, Kyushu Universiy, Kasuga, Fukuoka, Japan, November 15-17, 2004 Article No. 9 HIROTA RyogoTAKAHASHI Daisuke Received February 23, 2005 Research Institute for Applied Mechanics Kyushu University April, 2005

(Ryogo Hirota), (Daisuke Takahashi) 1 u m+1 1 n+1 u m n = δ(u m+1 n u m n+1) a) b) implicit KdV, KP explicit 1. 2. 3. 4. m, n 1 u m+1 1 n+1 u m n = δ(u m+1 n u m n+1) 1

δ BBB (Back ultradiscretized Box and Ball system) Satoshi Tujimoto and Ryogo Hirota, Ultradiscrete KdV Equations, J. Phys. Soc.Jpn. 67(1998) pp.1809-1810 Discrete KdV equation 1 1 u m+1 n u m n = δ(u m+1 n 1 u m n+1) m + n n BBB 1 u m+1 1 n+1 u m n = δ(u m+1 n u m n+1) u m n = f n m+1 fn+1 m fn m f m+1 n+1 fn+1 m+1 fn m 1 δfn m+1 f m 1 n+1 c 0 f m n+1f m n = 0 c 0 c 0 = 1 δ f m n f m n g(n), g(n) n Discrete KdV m + n n n m f(m, n) = 1 + e ξ 1 + e ξ 2 + a 12 e ξ 1+ξ 2, e ξ j = r j p n j q m j, q j = 1 δp j, for j = 1, 2, p j δ a 12 = (p 1 p 2 ) 2 (p 1 p 2 1) 2 p j q j p j = 1 + δq j q j + δ, a 12 = (q 1 q 2 ) 2 (q 1 q 2 1) 2. 2

U m n U m n = F m+1 n + F m n+1 F m n F m+1 n+1, F m n = max(0, Ξ 1, Ξ 1, A 12 + Ξ 1 + Ξ 2 ), Ξ j = P j n Q j m c j, for j = 1, 2. P j = max(q j, 1) max(0, Q j 1), A 12 = Q 1 Q 2 Q 1 + Q 2. P = max(q, 1) max(0, Q 1) Q 1 = 2, Q 2 = 1 {0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0} {0,0,0,0,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0} {0,0,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0} {0,0,0,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0} {0,0,0,0,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,0,0} {0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0} {0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0} Q 1 := 5/2, Q 2 := 1 1/2 3

{0,0,1/2,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0} {0,0,0,0,0,1,1,1/2,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0} {0,0,0,0,0,0,0,1/2,1,1,0,0,0,1,0,0,0,0,0,0,0,0,0} {0,0,0,0,0,0,0,0,0,0,1,1,1/2,0,1,0,0,0,0,0,0,0,0} {0,0,0,0,0,0,0,0,0,0,0,0,1/2,1,0,1,1,0,0,0,0,0,0} {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,1,1/2,0,0,0} {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1/2,1,1,0} BBB u m+1 n = x n, u m n+1 + 1 δu m n = a n, 1 δ = b j BBB x j = a j b j+1 x j+1, for j = 0, 1, 2,. (x j+n = x j, j = 0, 1, ) x j BBB u m+1 j = u m j u m j 1 + N 1 k 1 { k s=1 δ(u m j s) 2 }u m j k 1 u m j + N 1 k 1 { k s=1 δ(u m j+1 s) 2 }u m j k or u m+1 n = 1 δu m n+1 ( 10) 4

{0,0,0,0,1,0,0,0,1,1} {1,1,0,0,0,1,0,0,0,0} {0,0,1,1,0,0,1,0,0,0} {0,0,0,0,1,1,0,1,0,0} {0,0,0,0,0,0,1,0,1,1} {1,1,0,0,0,0,0,1,0,0} {0,0,1,1,0,0,0,0,1,0} {0,0,0,0,1,1,0,0,0,1} {1,0,0,0,0,0,1,1,0,0} 4 6 {1,0,0,1,1,1,0,0,0,0} {0,1,0,0,0,0,1,1,1,0} {1,0,1,1,0,0,0,0,0,1} {0,1,0,0,1,1,1,0,0,0} {0,0,1,0,0,0,0,1,1,1} {1,1,0,1,1,0,0,0,0,0} {0,0,1,0,0,1,1,1,0,0} {1,0,0,1,0,0,0,0,1,1} {0,1,1,0,1,1,0,0,0,0} {0,0,0,1,0,0,1,1,1,0} 7 3 {1,1,1,0,0,1,1,0,1,1} {1,0,0,1,1,1,0,1,1,1} {0,1,1,1,1,0,1,1,1,0} {1,1,1,1,0,1,1,0,0,1} {1,1,1,0,1,0,0,1,1,1} {1,0,0,1,0,1,1,1,1,1} {0,1,1,0,1,1,1,1,1,0} {1,1,0,1,1,1,1,0,0,1} {1,0,1,1,1,0,0,1,1,1} {0,1,1,0,0,1,1,1,1,1} {1,0,1,1,1,1,1,1,1,1} {0,1,1,1,1,1,1,1,1,1} {1,1,1,1,1,1,1,1,1,0} {1,1,1,1,1,1,1,1,0,1} {1,1,1,1,1,1,1,0,1,1} {1,1,1,1,1,1,0,1,1,1} {1,1,1,1,1,0,1,1,1,1} {1,1,1,1,0,1,1,1,1,1} {1,1,1,0,1,1,1,1,1,1} 5

( 10) ( 12) (-1) (-2) {0,0,1,1,0,0,0,-1,0,0,1,0,0,0,0} {0,0,0,0,1,1,0,0,-1,0,0,1,0,0,0} {0,0,0,0,0,0,1,1,0,-1,0,0,1,0,0} {0,0,0,0,0,0,0,0,1,1,-1,0,0,1,0} {0,0,0,0,0,0,0,0,0,0,2,-1,0,0,1} {1,0,0,0,0,0,0,0,0,0,-1,2,0,0,0} {0,1,0,0,0,0,0,0,0,0,0,-1,1,1,0} {1,0,1,0,0,0,0,0,0,0,0,0,-1,0,1} {0,1,0,1,1,0,0,0,0,0,0,0,0,-1,0} {0,0,1,1,0,0,-2,0,0,1,0,0} {0,0,0,0,1,1,0,-2,0,0,1,0} {0,0,0,0,0,0,1,1,-2,0,0,1} {1,0,0,0,0,0,0,0,2,-2,0,0} {0,1,0,0,0,0,0,0,-1,3,-2,0} {0,0,1,0,0,0,0,0,0,-2,3,-1} {0,0,0,1,0,0,0,0,0,0,-2,2} {1,1,0,0,1,0,0,0,0,0,0,-2} {-2,0,1,1,0,1,0,0,0,0,0,0} 1. 1 u m+1 1 n+1 u m n = δ(u m+1 n u m n+1) 2. 3. Yes,! 6