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( ( (3 Lax : (4 Bäcklud : (5 (6 d q = e q q e q q + ( m q ( r = q q r ϕ(r ϕ (r 0 5 0 q q q + 5 3 4 5 m d q = ϕ (r + ϕ (r + ( Hooke ϕ(r = κr (κ > 0 ( d q = κ(q q + κ(q + q = κ(q + + q q (3 ϕ(r = a b e br + ar a b > 0 (4

( m d q = a [ e b(q q e b(q + q ] (5 m ab ( ( r 0 Hooke r ( α β t = αs q = βu α β (5 s u ( d R = e R + + e R e R (6 d log( + V = V + + V V (7 d log( + V = I I + (8 di = V V da = a (b b + (9 db = (a a R = q q + + V = e R I = dq a = e q q+ b = dq (0 (7 (8 LC * I I I + 入力 V V 出力 ( Hamiltoia H = m p + a e b(q q ( b * Ryogo Hirota ad Kimio Suzuki Studies o Lattice Solitos by Usig Electrical Circuit J Phys Soc Jp 8(970 366-367

Hamilto dq = H p dp = H q ( (5 ( N Poisso N Liouville-Arold * Liouville-Arold [0] N Hamilto Poisso N (i (ii (iii (iv (3 (Lax (8 = N I N+ = I V N+ = V LΨ = λψ L = I + V N + V I + V I 3 + V N I N + V N I N (3 I V t λ Ψ t Ψ dψ = BΨ B = 0 + V N + V 0 + V 0 + V N 0 + V N 0 (4 λ t (3 (4 L t L Lax dl = BL LB (5 (5 (8 Lax Tr L k (k = N *3 d Tr Lk = 0 k = N * (Kepler Lagrage Euler Kowalevskaya Kowalevskaya 889 (967 70 *3 q Tr L Tr L

A = (a i j B = (b i j Tr A = A B t d Tr AB = d i= k= a ik b ki = a ii Tr AB = i= i= k= i= k= a ik b ki = k= i= ( ( a da ik b ki + a ik b ki = Tr B + AdB b ki a ik = Tr BA Tr L k L = BL LB d Tr Lk = Tr ( L L k + LL L k + + L k L = Tr [ (BL LBL k + L(BL LBL k + + L k (BL LB ] = Tr [ (BL k LBL k + (LBL k LBL k + + (L k BL L k B ] = Tr ( BL k L k B = 0 (4 Bäcklud λ α t q q dq dq = λe q q + λ eq q + α = λe q q + λ eq q + + α (6 q q ( q q ( ( q (q (6 q (q (6 Bäcklud q = 0 λ = e κ α = (e κ + e κ (6 e q = X e κ X = e κ X (e κ + e κ X = e κ X (e κ + e κ X + e κ (7 (7 Riccati ( X = e q + e κ( +βt = β = sih κ = eκ e κ (8 + e κ+βt Bäcklud Bäcklud [4] 3 (7 (8 3 (8 e q = τ τ q = log τ τ (9

τ (9 ( d log τ τ τ = d τ log τ τ τ + (0 τ f (t τ τ τ (τ = τ τ + f (t τ ( f (x t g(x t D-operator D m x D t f g D m x D t f g = ( x x m ( t t f (x tg(x t x=x t=t ( D x f g = f x g f g x D x f g = f xx g f x g x + f g xx D x D t f g = f xt g f x g t f t g x + f g xt (3 ( + Leibitz ( D t τ τ = τ + τ f (t τ (4 ( (4 (biliear equatio (biliear form 4 ( ( D m x D t (a f + bg h = ad m x D t f h + bd m x D t g h (a b: ( D m x D t f g = ( m+ D m x D t g f (3 D m x D t f = m x t f (4 D m x D t e p x+q t e p x+q t = (p p m (q q e (p +p x+(q +q t 4 (4 [3] τ *4 ( ( q = 0 τ = ( f (t f (t = ( τ = ϵ τ = + ϵ f ( + ϵ f ( + ϵ 3 f (3 + (5 (4 ϵ *5 *4 *5

(5 (4 ϵ O(ϵ : f ( = f ( + + f ( f ( (6 O(ϵ : f ( f ( + f ( + f ( = D t f ( f ( + f ( + f ( f ( (7 O(ϵ 3 : f (3 f (3 + f (3 + f (3 = D t f ( f ( + f ( + f ( + f ( + f ( f ( f ( (8 f ( (6 (7 f ( = e η η = p + q t + η 0 p q η 0 : (9 q = ep + e p = (e p e p q = ± sih p D t f ( f ( + f ( + f ( f ( = D t e η e η + e p e η e p e η e η = 0 (4 0 (7 f ( f ( f ( + f ( + f ( = 0 = 0 f (k *6 (5 = 0 (k = 3 4 τ = + e η η = p ± sih p t + η 0 p η 0 : (30 (4 (30 (9 (8 - f ( (6 f ( (6 (7 f ( = e η + e η η i = p i + q i t + η i0 (i = (3 f ( f ( + f ( + f ( = D t f ( f ( + f ( + f ( f ( q i = ± sih p i (i = (3 = D t (eη + e η (e η + e η + ( e η +p + e η +p ( e η p + e η p (e η + e η = D t e η e η + e η +η +p p + e η +η p +p e η +η ( = (q q e η +η + e p p e p p e η +η = ( ( e p e p p ( e e p e p p 4 e p p 4 e η +η f ( = A e η +η (A : ( A (q + q e η +η A e p +p e p +p ( p e η +η ( = A e e p p ( e e p e p +p 4 e p +p 4 e η +η *6

A = e p p 4 e p p 4 e p +p 4 e p +p 4 = sih p p 4 sih p +p (33 4 (8 D t f ( f ( + f ( + f ( + f ( + f ( f ( f ( = D t (e η + e η A e η +η + ( e η +p + e η +p A e η +η p p + ( e η p + e η p A e η +η +p +p (e η + e η A e η +η (4 D t (e η + e η e η +η = D t e η e η +η + D t e η e η +η = [ q (q + q ] e η +η + [ q (q + q ] e η +η = q eη +η + q eη +η (8 A [ q eη +η q eη +η + ( e p e p e η +η + ( e p e p e η +η ] = 0 (8 f (3 f (3 f (3 + f (3 + f (3 = 0 = 0 f (k = 0 (k = 4 5 τ = + e η + e η + A e η +η η i = p i + q i t + η i0 q i = ± sih p i (i = A = sih p p 4 sih p +p 4 (34 (4 - f ( N N N- - (34 (30 - q = log + e p e η + e η η = p + q t q = sih p (35 q η 0 = 0 η = p ( v t v = p sih p v p > 0 t η q log = 0 η q log e p = p q p v ( 3 - (34 p > p > 0 η i = p i ( v i t v i = p i sih p i v i p i v > v > 0 ( v η = p ( v t = p ξ ξ = η = p ( v t = p ( v t + p (v v t = p ξ + p (v v t

4 4 3 3 - τ = + e η + e η + A e η +η = + e p ξ + e p ξ +p (v v t + A e (p +p ξ +p (v v t v v > 0 t ± t : τ + e p ξ = + e η t + : τ e p ξ +p (v v t + A e (p +p ξ +p (v v t = e η + A e η +η = e η ( + A e η + A e η *7 t ± v log A ( v η = p ( v t = p ξ ξ = η = p ( v t = p ( v t + p (v v t = p ξ + p (v v t τ = + e η + e η + A e η +η = + e p ξ +p (v v t + e p ξ + A e (p +p ξ +p (v v t t : τ e p ξ +p (v v t + A e (p +p ξ +p (v v t = e η + A e η +η = e η ( + A e η + A e η t + : τ + e p ξ = + e η t ± v log A (34 v v 4 r = q q t *7 τ (4 e P+Qt τ (4 q = log τ τ log ep( +Qt τ e P+Qt = q τ P τ e P+Qt τ τ e P+Qt τ

4 R = q q + - ( ( : τ (3 = Plücker (4 ( ( f (x y q x y = eq q e q q + ( x y r x y = er + + e r e r ( x y log( + V = V + + V V (3 x log( + V = I I + I y = V V (4 D xd y τ τ = τ + τ f (x y τ (5 q = log τ r = q q + = log( + V = log τ +τ I = q τ x = x log τ (6 τ ( ( t = x + y s = x y q s = 0 ( q x y = eq q e q q + t s τ q = e q q e q q + d q = e q q e q q + ( q = 0 τ = (5 f (x y = D xd y τ τ = τ + τ τ (7

τ τ = + ϵ f ( + ϵ f ( + ϵ 3 f (3 + (8 ϵ O(ϵ : x y f ( = f ( + + f ( f ( O(ϵ : x y f ( f ( + f ( + f ( = D xd y f ( f ( + f ( + f ( f ( O(ϵ 3 : x y f (3 f (3 + f (3 + f (3 = D x D y f ( f ( + f ( + f ( + f ( + f ( f ( f ( ( f ( = R ep x+q y = e ζ ζ = log R + P x + Q y + ζ 0 (P Q R ζ 0 - τ = + R ep x+q y P Q = ( R R (9 ( f ( = e ζ + e ζ ζ i = log R i + P i x + Q i y + ζ i0 (i = - τ = + e ζ + e ζ + A e ζ +ζ P i Q i = ( R i R i (i = 3- A = [ (P P (Q Q ( R R R ] R [ (P + P (Q + Q ( R R ] R R (0 τ = + e ζ + e ζ + e ζ 3 + A e ζ +ζ +A 3 e ζ +ζ 3 + A 3 e ζ +ζ 3 + A 3 e ζ +ζ +ζ 3 ( ( ζ i = log R i + P i x + Q i y + ζ i0 P i Q i = R i ( R i [ ( ] R (P i P j (Q i Q j i R j R j R i A i j = [ ( ] A 3 = A A 3A 3 (3 (P i + P j (Q i + Q j R i R j R i R j 3- A 3 Casorati - (0 P i = p i q i Q i = p i + q i R i = ( pi q i (4 P i Q i = ( R i R i A i j = (p i q i (p j q j (p i q j (p j q i (5 - (0 τ = + e η ξ + e η ξ + A e η +η ξ ξ η i = log p i + p i x y p i + η 0i ξ i = log q i + q i x y q i + ξ 0i A = (p q (p q (p q (p q (6

- (6 τ = + e η ξ + e η ξ + A e η +η ξ ξ f ( f ( + f ( f ( + (7 f (i = e η i + e ξ i η i = log p i + p i x y p i + η 0i ξ i = log q i + q i x y q i + ξ 0i (8 f ( f ( + f ( f ( = ( ( ( ( e η + e ξ p e η + q e ξ e η + e ξ p e η + q e ξ + = (p p e η +η + (q p e η +ξ + (p q e η +ξ + (q q e ξ +ξ + q p q q e η ξ + p q q q e η ξ + p p q q e η +η ξ ξ (9 (η i0 ξ i0 (9 (q p e η = e η +log(q p = e η (p q e η = e η +log(p q = e η (q q e ξ = e ξ +log(q q = e ξ (q q e ξ = e ξ +log(q q = e ξ + e η ξ + e η ξ + p p (q q e η + η ξ ξ = + e η ξ + e η ξ + (p p (q q e η + η ξ ξ q q (q p (p q (q p (p q η i ξ i η i ξ i *8 (7 (7 Casorati Wroski Wroski (7 f (i x = f (i + f (i y = f (k (0 N N (7 N N N Casorati τ = f ( f ( + f ( +N f ( f ( + f ( +N f (N f (N + f (N +N (7 f (i f (i x = f (i + f (i y ( (i = N = f (i ( *8 η i0 ξ i0

f (i = p i exp ( p i x y p i + η i0 + q i exp ( q i x y q i + ξ i0 (3 N- 3 Plücker (7 Plücker Plücker Grassma ( τ ( (7 Plücker Freema-Nimmo τ = f ( f ( + f ( +N f ( f ( + f ( +N f (N f (N + f (N +N = 0 N N j = f ( + j f ( + j f (N + j (4 τ = 0 N N x τ = 0 N N τ + = N N N y τ = N N (5 τ = 0 N ( x y + τ = N N (5 3 3 x f ( f ( + f ( +N f ( f ( + f ( +N f (N f (N + f (N +N = j= f ( + j f ( + j f ( + j+ f ( + j f ( + j f ( + j+ f (N + j f (N + j f (N + j+ (6 f (i x = f (i + x τ = 0 N N + 0 N N + 0 N N = N N + + 0 N N + 0 N N = 0 N N

f (i y = f (i y τ = 0 N N + 0 N N + + 0 N N = N N 0 0 N N 0 N N = N N x y τ = N N N N + + N N = 0 N N N N = τ N N Plücker (5 (7 0 = 0 N + 0 N N 0 N N N N N N N N N s (7 (7 N 4 0 N N( (7 Plücker (7 Laplace 3 ( Laplace A = (a i j i j N N N A i i i l j j j l A i i l j j l l l A i i i l j j j l A i i l j j l (N l (N l l i i l i < < i l N A = j < < j l N ( i + +i l + j + + j l A i i i l j j j l A i i i l j j j l (8 l = i = (8 [5] 0 N N Ø 0 0 N Ø N N 0 = Ø N N N (9 } {{ }} {{ } N N 0 ( Ø 0 N ( (N Ø Ø = Ø N N N

Ø 0 N Ø Ø Ø = Ø N N N } {{ }} {{ } N N l = N i = i N = N Laplace N N 0 N + 0 0 (9 Laplace (7 ( Plücker a 0 N N b 4 ( τ x j y j ( j = f (i x j = f (i + j f (i y j = f (i j (30 τ [??] (3 Plücker (hierarchy x j ( y j KP τ ( ( τ Plücker Plücker 4 Casorati τ f (x y x f (x y x f (x y y f (x y x y f (x y x y f (x y τ = y f (x y x y f (x y x y f (x y (3

f (x y τ D xd y τ τ = τ + τ ( = 0 (3 ( τ = 0 τ 0 = τ = f (33 (3 x y Wroskia Wroskia (33 q = log τ τ : q 0 = R = log τ +τ τ : R 0 = (34 (3 Plücker (3 Freema-Nimmo f (x y x f (x y x f (x y τ + = = 0 j = y f (x y x y f (x y x y f (x y y f (x y x y f (x y x y f (x y x j f (x y x j y f (x y x j y f (x y (35 ( + ( + j ( + j x f j ( + ( + 0 = 0 Ø ϕ ϕ Ø 0 ϕ ϕ ϕ = 0 0 ϕ = ϕ ϕ 0 Laplace 0 = 0 0 ϕ + 0 ϕ 0 ϕ ϕ 0 ϕ 0 ϕ 0 0 (36 (37 Plücker τ f x f 0 0 0 ϕ ϕ = y f x y f 0 0 = y f x y f 0 y f x y f 0 f x f y f x y f = τ (38

( + f x f 0 f x f 0 ϕ = y f x f 0 = y f x y f y f x f y f x y f x y f y f 0 = y τ (39 ( + 0 ϕ = y f x f x f x 0 y f x y f 0 y f x y f x y f = f x f x y f x y f x y f = x τ (40 0 ϕ = f x f x 0 y f x y f x f 0 y f x y f x y f y f x y f x y f 0 = f x f x y f x y f x f y f x y f x y = x y τ (4 Plücker (37 ( x y τ τ ( x τ ( y τ = τ+ τ (4 (3 ( X i (x Y i (y (i = N f (x y = X i (xy i (y (43 i= τ N τ N = Y Y N y Y y Y N y Y y N N Y N X x X N x X X N x X N N x X N = Y(y X(x (3 τ N+ = 0 (44

*9 = N ( log h x y = h + + h h h = τ τ + τ 889 Darboux ( [7] Darboux (3 f (t f (t t f (t τ = (45 t f (t t f (t ( t τ τ ( t τ = τ + τ τ = 0 τ 0 = τ = f (t (46 f (t = V I LC i= e λ it+µ i V = τ τ + I τ = d log τ (47 τ d log V = I I + di = V V (48 * 0 V 0 = 0 V N = 0 (49 V 0 Lax L (3 t [8] q (reductio ( ( q s = 0 t = x + y s = x y (50 q + = q (5 *9 Biet-Cauchy [5] (3 *0 (8 + V V

( q 0 x y = eq q 0 e q 0 q q x y = eq 0 q e q q 0 (5 v x y = ( e v e v v = q 0 q (53 v = 4 sih v (54 x y v = iθ ir v = 4 si v (55 x y (54 (55 sih-gordo sie-gordo Casorati ( (3 (50 τ f (i f (k q s = s log τ = sτ sτ = 0 τ s s [ = p k ep k x y p k + q k eq k x y q k = p k exp s f (k = s f (k s τ = cost τ (56 (p k pk t + ] [ (p k + pk s + q k (q exp k qk t + ] (q k + qk s [ (p k + pk p k (p exp k pk t + ] (p k + pk s + [ (q k + qk q k (q exp k qk t + ] (q k + qk s = cost f (i (56 p k + p k = q k + q k s f (k = (p k + pk s τ = C N τ C N = q k = p k (57 f (k (58 i= (p i + pi (59 (56 s t (59 (7 (4( f (t =

4 ( N- τ = q = log τ τ (60 f ( f ( + f ( +N f (N f (N + f (N +N (6 f (k = p k e t (p k pk +η k0 + p t k e (p k pk +ξ k0 (6 sih-gordo (5 τ N- ( (3 τ + = cost τ (63 f (k + = p k p k ep k x y p k + q k q k eq k x y q k f (k p k = q k q k = p k (64 (7 v f (k + = p k f (k (65 N τ + = λ τ λ = p i (66 i= D xd y τ 0 τ 0 = λ τ τ 0 D xd y τ τ = λτ 0 τ (67 v = q 0 q = log τ log λ (68 τ 0 5 sih-gordo (54 N- f (k τ = v = log τ τ 0 log λ (69 f ( f ( + f ( +N f (N f (N + f (N +N ( N = p k ep k x y p +η k0 k + ( p k e p k x y p +ξ k k0 λ = (70 i= p i (7

sie-gordo (55 Casorati η k0 R ξ k0 = πi (7 Gram 3 Bäcklud ( Bäcklud ( Bäcklud Lax (3 Bäcklud 3 Bäcklud Bäcklud Bäcklud Bäcklud 3 ( Bäcklud τ λ λ λ 3 τ D t τ τ = τ + τ τ (73 D t τ τ = λ τ + τ λ τ τ D t τ + τ = λ τ τ + + λ 3 τ + τ (74 τ (73 τ (73 τ (74 τ (73 τ (73 τ (74 P = 0 τ (73 6 ( x y τ τ ( [ Dx D y τ τ ] τ (τ [ D x D y τ τ ] = Dx ( Dy τ τ τ τ (75 ( D x (τ + τ (τ τ = [ D x τ + τ ] τ τ + τ + τ [ Dx τ τ ] (76 [3] P [ ] [ ] P = D t τ τ τ + τ + τ (τ (τ D t τ τ τ + τ + (τ [ ] = D t Dt τ τ τ τ + τ τ + τ τ + τ τ τ τ + = D t [ Dt τ τ λ τ + τ ] τ τ + λ [ Dt τ + τ + λ τ τ + ] τ τ + λ [ Dt τ τ + λ τ τ ] τ τ +

(75 3 (76 D t f f = 0 D t τ τ λ τ + τ = λ τ τ (77 0 3 0 D t τ + τ = τ τ + + λ 3 τ + τ (78 λ λ 3 3 (74 P = 0 Bäcklud (74 q (74 τ τ τ + τ (80 (8 q = log τ τ q = log τ τ (79 ( log τ ( log τ τ + τ = λ λ τ τ (80 ( log τ+ ( log τ = τ τ + + λ 3 λ τ + τ (8 ( log τ τ τ = λ + τ τ λ + λ 3 (8 τ τ τ λ τ τ (80 (8 ( log τ τ + τ = λ + τ τ λ + λ 3 (83 τ τ τ λ τ τ q q dq dq = λ e q +q + λ e q q λ + λ 3 = λ e q ++q + λ e q q λ + λ 3 (84 Bäcklud (6 3 Bäcklud Lax τ Bäcklud (74 Lax * τ = τ Ψ + (85 (74 (85 Ψ + = λ τ + τ Ψ τ λ Ψ + ( Ψ + + log τ + Ψ + = Ψ + + λ 3 Ψ + τ λ (86 * Bäcklud τ

(6 V I Ψ = λ ( + V Ψ + λ Ψ Ψ = (I + λ 3 Ψ + Ψ + λ λ ( + V Ψ I Ψ + Ψ + = (λ + λ 3 Ψ λ Ψ = λ ( + V Ψ + λ Ψ (87 (88 ( λ e λ t Ψ Ψ λ 3 = λ ( + V Ψ + I Ψ + Ψ + = λψ Ψ = ( + V Ψ (89 Lax (3 (4 33 Bäcklud Bäcklud 4 ( Bäcklud τ λ λ λ 3 τ D xd y τ τ = τ + τ τ (90 D y τ τ = λ τ + τ λ τ τ D x τ + τ = λ τ τ + + λ 3 τ + τ (9 τ (90 τ (90 τ (74 τ (90 ( 4 ( Bäcklud (6 τ N N Casorati (N- ( ( τ (N + (N + Casorati (N + - * ( N N Casorati τ τ N (7 (9 Laplace Laplace 0 N Ø N ϕ 0 = Ø N N ϕ ϕ = 0 0 (9 0 = 0 N N N ϕ + 0 N N N ϕ 0 N ϕ N N (93 * τ

( ϕ N τ N = τ τ N+ 0 = τ N τn + + τn ( y τ N τ N ( y τ N (94 = τ (9 λ = λ = 0 D y τ τ = τ + τ (95 3 τ N = τ τ N+ = τ (9 ( (9 Laplace ( Lax τ = τ Ψ y Ψ = λ ( + V Ψ + λ Ψ x Ψ = (I + λ 3 Ψ + λ Ψ + (96 x ( y Ψ = y ( x Ψ (4 ( Bäcklud t = x + y s = x y s sih-gordo sie-gordo sie-gordo Bäcklud θ θ y θ + θ = λ si θ + θ = si θ θ λ x (97 Bäcklud 875 4 (97 θ( θ θ( θ sie-gordo (55 3 τ KP [3] [] Fermio Boso Date-Jimbo-Kashiwara-Miwa [] [] [4] [7] [5] [9] [] [6] [6]

* 3 * 4 [] MJ H (99 [] (007 [3] (99 [4] (003 [5] (009 [6] (00 [7] (00 [8] (006 [9] (000 [0] (994 [] (009 [] Y Ohta J Satsuma D Takahashi T Tokihiro A elemetary itroductio to Sato Theory Progr Theor Phys Suppl 94(988 0 4 [3] (00 [4] (987 [5] (00 [6] C Rogers ad WK Schief Bäcklud ad Darboux Trasformatios Geometry ad Moder Applicatios i Solito Theory (Cambridge Uiversity Press Cambridge 00 *3 *4 kaji@imikyushu-uacjp

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