t, x (4) 3 u(t, x) + 6u(t, x) u(t, x) + u(t, x) = 0 t x x3 ( u x = u x (4) u t + 6uu x + u xxx = 0 ) ( ): ( ) (2) Riccati ( ) ( ) ( ) 2 (1) : f

Similar documents

Korteweg-de Vries

163 KdV KP Lax pair L, B L L L 1/2 W 1 LW = ( / x W t 1, t 2, t 3, ψ t n ψ/ t n = B nψ (KdV B n = L n/2 KP B n = L n KdV KP Lax W Lax τ KP L ψ τ τ Cha

5. [1 ] 1 [], u(x, t) t c u(x, t) x (5.3) ξ x + ct, η x ct (5.4),u(x, t) ξ, η u(ξ, η), ξ t,, ( u(ξ,η) ξ η u(x, t) t ) u(x, t) { ( u(ξ, η) c t ξ ξ { (

#A A A F, F d F P + F P = d P F, F y P F F x A.1 ( α, 0), (α, 0) α > 0) (x, y) (x + α) 2 + y 2, (x α) 2 + y 2 d (x + α)2 + y 2 + (x α) 2 + y 2 =

²ÄÀÑÊ¬Î¥»¶ÈóÀþ·¿¥·¥å¥ì¡¼¥Ç¥£¥ó¥¬¡¼ÊýÄø¼°¤ÎÁ²¶á²òÀÏ Asymptotic analysis for the integrable discrete nonlinear Schrödinger equation

W u = u(x, t) u tt = a 2 u xx, a > 0 (1) D := {(x, t) : 0 x l, t 0} u (0, t) = 0, u (l, t) = 0, t 0 (2)

(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y

構造と連続体の力学基礎

Note.tex 2008/09/19( )

Introduction to Numerical Analysis of Differential Equations Naoya Enomoto (Kyoto.univ.Dept.Science(math))

No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2

u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT

2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =

S I. dy fx x fx y fx + C 3 C dy fx 4 x, y dy v C xt y C v e kt k > xt yt gt [ v dt dt v e kt xt v e kt + C k x v + C C k xt v k 3 r r + dr e kt S dt d

S I. dy fx x fx y fx + C 3 C vt dy fx 4 x, y dy yt gt + Ct + C dt v e kt xt v e kt + C k x v k + C C xt v k 3 r r + dr e kt S Sr πr dt d v } dt k e kt

e a b a b b a a a 1 a a 1 = a 1 a = e G G G : x ( x =, 8, 1 ) x 1,, 60 θ, ϕ ψ θ G G H H G x. n n 1 n 1 n σ = (σ 1, σ,..., σ N ) i σ i i n S n n = 1,,

V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H

Black-Scholes [1] Nelson [2] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [2][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-W

II A A441 : October 02, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka )

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

(1) (2) (3) (4) HB B ( ) (5) (6) (7) 40 (8) (9) (10)

微分積分サンプルページこの本の定価判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.

II Karel Švadlenka * [1] 1.1* 5 23 m d2 x dt 2 = cdx kx + mg dt. c, g, k, m 1.2* u = au + bv v = cu + dv v u a, b, c, d R

( ) ( 40 )+( 60 ) Schrödinger 3. (a) (b) (c) yoshioka/education-09.html pdf 1

2 1 1 (1) 1 (2) (3) Lax : (4) Bäcklund : (5) (6) 1.1 d 2 q n dt 2 = e q n 1 q n e q n q n+1 (1.1) 1 m q n n ( ) r n = q n q n 1 r ϕ(r) ϕ (r)

9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P

1 No.1 5 C 1 I III F 1 F 2 F 1 F 2 2 Φ 2 (t) = Φ 1 (t) Φ 1 (t t). = Φ 1(t) t = ( 1.5e 0.5t 2.4e 4t 2e 10t ) τ < 0 t > τ Φ 2 (t) < 0 lim t Φ 2 (t) = 0

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )

p = mv p x > h/4π λ = h p m v Ψ 2 Ψ

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google

Microsoft Word - 信号処理3.doc

2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n

O x y z O ( O ) O (O ) 3 x y z O O x v t = t = 0 ( 1 ) O t = 0 c t r = ct P (x, y, z) r 2 = x 2 + y 2 + z 2 (t, x, y, z) (ct) 2 x 2 y 2 z 2 = 0

() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)

x A Aω ẋ ẋ 2 + ω 2 x 2 = ω 2 A 2. (ẋ, ωx) ζ ẋ + iωx ζ ζ dζ = ẍ + iωẋ = ẍ + iω(ζ iωx) dt dζ dt iωζ = ẍ + ω2 x (2.1) ζ ζ = Aωe iωt = Aω cos ωt + iaω sin

x (x, ) x y (, y) iy x y z = x + iy (x, y) (r, θ) r = x + y, θ = tan ( y ), π < θ π x r = z, θ = arg z z = x + iy = r cos θ + ir sin θ = r(cos θ + i s

1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1

I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%

2009 IA I 22, 23, 24, 25, 26, a h f(x) x x a h

meiji_resume_1.PDF

d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r

July 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i

f(x) = f(x ) + α(x)(x x ) α(x) x = x. x = f (y), x = f (y ) y = f f (y) = f f (y ) + α(f (y))(f (y) f (y )) f (y) = f (y ) + α(f (y)) (y y ) ( (2) ) f

9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x

() Remrk I = [0, ] [x i, x i ]. (x : ) f(x) = 0 (x : ) ξ i, (f) = f(ξ i )(x i x i ) = (x i x i ) = ξ i, (f) = f(ξ i )(x i x i ) = 0 (f) 0.

: , 2.0, 3.0, 2.0, (%) ( 2.

x i [, b], (i 0, 1, 2,, n),, [, b], [, b] [x 0, x 1 ] [x 1, x 2 ] [x n 1, x n ] ( 2 ). x 0 x 1 x 2 x 3 x n 1 x n b 2: [, b].,, (1) x 0, x 1, x 2,, x n

1 nakayama/print/ Def (Definition ) Thm (Theorem ) Prop (Proposition ) Lem (Lemma ) Cor (Corollary ) 1. (1) A, B (2) ABC

M3 x y f(x, y) (= x) (= y) x + y f(x, y) = x + y + *. f(x, y) π y f(x, y) x f(x + x, y) f(x, y) lim x x () f(x,y) x 3 -

9 5 ( α+ ) = (α + ) α (log ) = α d = α C d = log + C C 5. () d = 4 d = C = C = 3 + C 3 () d = d = C = C = 3 + C 3 =

7. y fx, z gy z gfx dz dx dz dy dy dx. g f a g bf a b fa 7., chain ule Ω, D R n, R m a Ω, f : Ω R m, g : D R l, fω D, b fa, f a g b g f a g f a g bf a

2012 IA 8 I p.3, 2 p.19, 3 p.19, 4 p.22, 5 p.27, 6 p.27, 7 p

() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (

(5) 75 (a) (b) ( 1 ) v ( 1 ) E E 1 v (a) ( 1 ) x E E (b) (a) (b)

(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)

2.2 ( y = y(x ( (x 0, y 0 y (x 0 (y 0 = y(x 0 y = y(x ( y (x 0 = F (x 0, y(x 0 = F (x 0, y 0 (x 0, y 0 ( (x 0, y 0 F (x 0, y 0 xy (x, y (, F (x, y ( (

Grushin 2MA16039T

t (x(t), y(t)), a t b (x(a), y(a)) t ( ) ( ) dy s + dt dt dt [a, b] a a t < t 1 < < t n b {(x(t i ), y(t i ))} n i ( s(t) ds ) ( ) dy dt + dt dt ( ) d

24 I ( ) 1. R 3 (i) C : x 2 + y 2 1 = 0 (ii) C : y = ± 1 x 2 ( 1 x 1) (iii) C : x = cos t, y = sin t (0 t 2π) 1.1. γ : [a, b] R n ; t γ(t) = (x

Transcription:

: ( ) 2008 5 31 1 f(t) t (1) d 2 f(t) + f(t) = 0 dt2 f(t) = sin t f(t) = cos t (1) 1 (2) d dt f(t) + f(t)2 = 0 (1) (2) t (c ) (3) 2 2 u(t, x) c2 u(t, x) = 0 t2 x2 1 (1) (1) 1

t, x (4) 3 u(t, x) + 6u(t, x) u(t, x) + u(t, x) = 0 t x x3 ( u x = u x (4) u t + 6uu x + u xxx = 0 ) ( ): ( ) (2) Riccati ( ) 2008 4 ( ) ( ) 2 (1) : f 1 (t) f 2 (t) f(t) = a 1 f 1 (t) + a 2 f 2 (t) (a 1, a 2 ) f (t) a 1 f 1(t) + a 2 f 2(t) f(t) (1) ( ; ) (1) sin t cos t f(t) = a sin t + b cos t (a, b ) 2 2

( ) 3! 2 2.1 8 9 30 1 1.5 1834 John Scott-Russel ( - ) 4 Union Canal 1995 http://www.ma.hw.ac.uk/solitons/press.html Scott-Russel (solitary wave; wave of translation ) ( ) 1882 Boussinesq ( ) 1895 Korteweg ( ) de Vries ( ) Newton ( ) 17 18 Euler ( ) Lagrange ( ) 19 Boussinesq Korteweg-de Vries Boussinesq Korteweg-de Vries 3 4 http://www.ma.hw.ac.uk/~chris/scott russell.html 3

u(x, t) x ( ) t u(x, t) 1 (3) c f(x) u(x, t) = f(x ct) u(x, t) = f(x + ct) (3) x c Scott-Russel u(x, t) Boussinesq Korteweg-de Vries Korteweg-de Vries ( KdV ) 1 (4) ( x, t u(x, t) ) KdV (k, δ ) (5) u(x, t) = 2k 2 sech 2( k(x ct) + δ ), c = 4k 2 sech x e x 2 sech x = sech x x = 0 1 x ± 0 e x + e x u(x, t) 2k 2 1 x t x ct c = 4k 2 x 5 KdV 4k 2 2k 2 x=ct x 1: KdV KdV Korteweg de Vries 5 Boussinesq Scott-Russel 4

2.2 KdV ( ) KdV ( ) E. Fermi ( ; ) 1 ( ) Fermi Pasta ( ), Ulam ( ) N. Zabusky ( ) M. D. Kruskal ( ) KdV (t = 0 u(x, t)) ( u(x, 0) = sin x) u(x, t) http://www.math.h.kyoto-u.ac.jp/~takasaki/soliton-lab/gallery/solitons/ ( 2, 3, 4 ) Korteweg-de Vries (solitary wave) -on ( electron, photon, proton ) Solitary wave + -on = SOLITON 5

2: KdV ; 3: KdV ; 6 2.3 Boussinesq : u(x, t) (6) 3 4 u tt + u xx + 1 4 u xxxx + 3 2 (uu x) x = 0. KdV : v(x, t) (7) v t + 6v 2 v x + v xxx = 0. KdV (4) KdV 6 solitron 6

KP 7 : u(x, y, t) (8) (u t + 6uu x + u xxx ) x + 3u yy = 0. KdV (4) KdV u(x, t) y (x, y, t) KP KP y 2 KdV Sine-Gordon : ( ) u(x, t) (9) u tt u xx + sin u = 0. sin u m 2 u m Klein-Gordon sine-gordon ( 4) x Sine-Gordon u(x, t) 4: Sine-Gordon 2- : t n ( ) u n (t) (10) u tt = e u n 1 u n e u n u n+1. [To1] 3 2.1 2.2 made in Japan 2.1 ( 4.1 ) ( Novikov ) ( 5.2 ) 7 Kadomtsev ( )-Petviashvili ( ) 7

3.1 Lax 1 Riccati (2) (11) f(t) = g (t) g(t) f(t) = d log g(t) f(t) g(t) g(t) = dt e R f(t) dt Riccati (2) (12) d 2 dt 2 g(t) = 0 g(t) = a 1 t + a 0 (a 0, a 1 ) Riccati (13) f(t) = a 1 a 1 t + a 0 KdV u xxx u xx Burgers u w x /w w KdV 1967 Miura ( ; ) w x /w w xx /w u = w xx /w w v = iw x /w (i = 1), (14) w xx + uw = 0, iw x + vw = 0 v v KdV (7) Gardener ( ), Greene ( ), Kruskal, Miura (GGKM ) w(x, t; λ) : (15) 2 x 2 w(x, t; λ) + u(x, t)w(x, t; λ) = λ2 w(x, t; λ) λ x t (15) ( ) Schrödinger ( ) ( t ; t ) u(x) w(x; λ) ( ) u(x) ( ) w(x; λ) ( u(x) w(x; λ) ) (16) x w(x; λ) = 1 a(λ) e iλx, x + w(x; λ) = e iλx + r(λ)e iλx 8

(a(λ), r(λ) λ ) e iλx e iλx λ w(x; λ) λ u(x) 1/ a(λ) r(λ) ( 5) 1/a(λ) r(λ) 8 u(x) e -i x e -i x u(x) e i x x 5: u(x) u(x) u(x) ( λ ) u(x) u(x)? Gelfand-Levitan(-Marchenko) ( ) u(x)! (M. Kac ( ) ) Schrödinger (15) KdV GGKM KdV u(x, t) Schrödinger (15) λ t ( ) t 1/a(λ) t r(λ, t) t r(λ, t) = r(λ, 0)e 8iλ3 t u(x, t) t = 0 u(x, 0) t 8 a(λ) λ c j, η j (j = 1,..., n; n u(x) ) r(λ) {r(λ), η 1,..., η n, c 1,..., c n } 9

u(x, t) u(x, 0) u(x, t) KdV! KdV (17) u(x, 0) a(λ), r(λ, 0) u(x, t) GLM a(λ), r(λ, t) = r(λ, 0)e 8iλ3 t r(λ, t) = 0 ( ) P. D. Lax ( ; 2007 ) λ Schrödinger (15) w(x, t; λ) L = 2 +u(x, t) x2 w λ ( ) x(t) w v(t) = y(t) ( ) a(t) b(t) L P (t) = v(t) c(t) d(t) P (t) P (t) λ t : (18) P (t)v(t) = λv(t) ( ) P (t) P (0) U(t) (19) P (t) = U(t)P (0)U(t) 1 P (t) U(t) U(t) A(t) ( ) d d (20) U(t) = A(t)U(t), A(t) = dt dt U(t) U(t) 1, (19) (21) d dt du(t) P (t) = P (0)U(t) 1 + U(t)P (0) d( U(t) 1) dt dt = du(t) U(t) 1 U(t)P (0)U(t) 1 + U(t)P (0) dt = A(t)P (t) P (t)a(t) ( U(t) ) 1 du(t) U(t) 1 dt 10

[A(t), P (t)] P (t) (22) d P (t) = [A(t), P (t)] dt L = 2 + u(x, t) ( x u(x, t) KdV L (23) t L = [A, L], A = 4 3 + 6u + 3u x, ) (L A 3 f(x) ) KdV (23) Lax L Lax (L, A) Lax (Lax pair) KdV KdV [TD] 3.2 KdV Sine-Gordon (RCA ) ([ ] ) (5) KdV Miura (14) KdV KdV f(x) : ( ) f f (f ) 2 f = = d2 log f. f 2 f dx2 f(x) = cosh x f (x) = sinh x, f (x) = cosh x 9 (5) (24) u(x, t) = 2 d2 dx 2 log cosh( k(x ct) + δ ) 2 ( 3 ) (1968 ; [To1] ) n ( n ) KdV Sine-Gordon ([ ] ): KdV f(x, t) (25) D x (D t + D 3 x)f f = 0 9 cosh x = ex + e x, sinh x = ex e x : 2 2 11

f(x, t) (26) u(x, t) = 2 2 log f(x, t) x2 u(x, t) KdV D : (27) D m x D n t f(x, t) g(x, t) := m x n t f(x + x, t + t )g(x x, t t ) x =t =0. x fg = f g + fg D x f g = f g fg, 2 xfg = f g + 2f g + fg D 2 xf g = f g 2f g + fg f (25) KdV v = x (log f) 1 f 2 D x(d t + D 3 x)f f = 2(v t + 6v 2 x + v xxx ) 0 0. x u = 2v x KdV KdV Sine-Gordon ( ) n (28) f(x, t) = 1 + εf (1) (x, t) + ε 2 f (2) (x, t) + u(x, t) 2 1 (28) f (1) 1 f (2) ( 3 ) f 4.1 ( ) (26) f ( ) [Hi] 12

4?? 1981 KP Grassmann 4.1 1970 Ising Lax (23) (t ) ( ): ( ) τ ([ ] ) ((25) D x, D t 4 ) ([ ] ) 10 ( ):? (1980 ) 420! ([ ] ) 10 n ; n = 3 3 = 2 + 1 = 1 + 1 + 1 3, n = 4 4 = 3 + 1 = 2 + 2 = 2 + 1 + 1 = 1 + 1 + 1 + 1 5. 13

( ) KdV Lax (23) Lax L 2 + u(x, t) (29) L = + u 2 1 + u 3 2 + = u n 1 n, (u 0 = 1, u 1 = 0) 1!?? n=0 Lax (23) KdV ( (22) ) n ( ) Lax [A, L] = AL LA ( )( f(x) m g(x) n) = m(m 1) (m r + 1) f(x)g (r) (x) m+n r r! r 0 m, n ( ) P = a n (x) n n P +, P (30) P + = n 0 a n (x) n, P = n<0 a n (x) n. 1970 Gelfand ( ), Dikii ( ), Manin ( ) Wilson ( ), Adler ( ) (29) Lax t 1, t 2,..., t n,... ( t ) u i x t u i (x, t) Lax (31) L t n = [B n, L], (n = 1, 2,... ). 14

B n L n B n = (L n ) + KP hierarchy, KP B n (32) B m t n B n t m + [B m, B n ] = 0, (m, n = 1, 2,... ) m = 2, n = 3 KP (8) (u 2 = u, t 1 = x, t 2 = y, t 3 = t, t n ; (8) ) KP Lax L (29) L 2 = 2 + 2u 2 + ( ) 0 KP u = 2u 2 KdV (t 1 = x, t 3 = t, t n ) KdV hierarchy, KdV ( 2 + u L ) L 2 L 3 Boussinesq KP Schrödinger (15) (33) (34) Lw(x, t; λ) = λw(x, t; λ), w(x, t; λ) = B n w(x, t; λ). t n (34) (33) L n (n = 1, 2,... ) w(x, t; λ) (35) w(x, t; λ) = (1 + w 1 (x, t)λ 1 + w 2 (x, t)λ 2 + )e xλ+ξ(t,λ), ξ(t, λ) = t n λ n, n e xλ+ξ(t,λ) = λ n e xλ+ξ(t,λ) (n 0 n ) (36) w(x, t; λ) = (1 + w 1 (x, t) 1 + w 2 (x, t) 2 + )e xλ+ξ(t,λ) n w(x, t; λ) Schrödinger (16) (35) 11 w(x, t; λ) τ(t) = τ(t 1, t 2,... ) ( ) τ t 1 + x λ 1, t 2 λ 2, t 2 3 λ 3,... 3 (37) w(x, t; λ) = e xλ+ξ(t,λ), τ(t 1 + x, t 2, t 3,... ) 11 (16) ±iλ (35) λ n=1 15

τ τ(t) f τ τ Schur ( ) Plücker ( ) KP Grassmann ( ) Grassmann ( ) Plücker Π ( 6) v = a 1 a 2 a 3 w = b 1 b 2 b 3 ( ) a 2 b 2 (38) αx + βy + γz = 0, α : β : γ = a 3 b 3 : a 1 b 1 : a 1 b 1 a 3 b 3 a 2 b 2 ( ) z v y w x 6: = GM(3; 2) (38) α : β : γ GM(3; 2) = {[α : β : γ] (α, β, γ) (0, 0, 0)} (α, β, γ 0 (38) ) 16

( (x, y, z, w) ) v = a 1 a 2 a 3 w = b 1 b 2 b 3 a 4 b 4 (39) 23 x 13 y + 12 z = 0, 24 x 14 y + 12 w = 0, 34 x 14 z + 13 w = 0, 34 y 24 z + 23 w = 0 12 ij a i b i (40) ij = = a ib j a j b i a j b j 12 : 13 : 14 : 23 : 24 : 34 GM(4; 2) GM(4; 2)? = {[ 12 : 13 : 14 : 23 : 24 : 34 ] ( 12,..., 34 ) (0, 0, 0, 0, 0, 0)} (39) (x, y, z, w) v, w (40) (41) 12 34 13 24 + 14 23 = 0, ( [Ha2] 12.4 ) ij ( 0 ) (39) { } ( 12,..., 34 ) (0,..., 0), (42) GM(4; 2) = [ 12 : : 34 ] 12 34 13 24 + 14 23 = 0 N m Grassmann GM(N; m) ( m ) (38) (39) N C m (N m ) Plücker Plücker (41) Plücker ( N, m ) GM( ; 2 ) Grassmann 13 Grassmann Grassmann Plücker 12 v, w 13 (..., x 2, x 1, x 0, x 1, x 2,... ) x 0 = x 1 = x 2 = = 0 17

(ξ λ ) λ: GM(4, 2) (42) Plücker ij (i, j) = (1, 2),..., (3, 4) (43) (ξ λ ) λ: = (ξ φ, ξ 1, ξ 2, ξ }{{ 1,1, ξ } 3, ξ 2,1, ξ 1,1,1, }{{} 2 3... ) ( φ 0 ) λ Schur t = (t 1, t 2, t 3,... ) s λ (t) ( [MJD] 9.3 ) Schur s φ (t) = 1, s 1 (t) = t 1, s 2 (t) = t2 1 2 + t 2, s 1,1 (t) = t2 1 2 t 2, s 3 (t) = t3 1 6 + t 1t 2 + t 3, s 2,1 (t) = t3 1 3 t 3, s 1,1,1 (t) = t3 1 6 t 1t 2 + t 3. λ n s λ (t) n t n n t = (t 1, t 2, t 3,... ) ( ) s λ (t) ( ) KP τ τ(t) Schur (44) τ(t) = λ: ξ λ s λ (t) (ξ λ ) λ: Plücker Grassmann (ξ λ ) λ Grassmann Plücker (44) KP τ KP Grassmann Schur ξ λ τ(t) Plücker τ(t) ( )! 1 Grassmann [SN] [SU] 18

4.2 Grassmann Grassmann KP hierarchy Grassmann KP Grassmann Grassmann GM(N; m) GM(N; m) N m GM(3; 2) 6 3 3 a 11 a 12 a 13 x a 11 x + a 12 y + a 13 z a 21 a 22 a 23 y = a 21 x + a 22 y + a 23 z a 31 a 32 a 33 z a 31 x + a 32 y + a 33 z ( ) GM(3; 2) (= ) GM(3; 2) 3 3 (GL(3) GL 3 ) GM(3; 2) GL(3) ( = ) Grassmann GM(N; m) N N GL(N) Grassmann GL( ) ( ) Lie ( ) ( Lie ) Lie Lie 1960 Lie ( ) ( KP ) GL( ) ( ) Grassmann KP KP GL( ) 14 n n + 1 τ(t) = 1 τ (= 0 ) 14 19

( ) Ising ( ) τ Lie Noether ( ) KP ( ) Liouville ( )-Arnold ( ) KP ( 1970 [W] ) [MJD] 5 1981 ( ) ( ) 5.1 [ ] 1970 ( Schrödinger ) 1980 [Ha1] ( ) 1970 20

([Sh]) 2000 5.2 KP ( ) GL( ) ( Lie ) 1980 Novikov : 3 ( ) KP (8) τ KP τ KP : U(1) KP τ Witten : KP ( KdV ) τ (45) Z N (t) = X: N e β tr (t 1 X+t 2 X 2 +t 3 X 3 + ) dx (46) Z N (Λ) = e tr (X 3 +ΛX) dx, X: N Λ : dx X N ( ) X N τ ( (46) t n = 1 n tr Λn KP ) (46) ( ) KdV τ Witten ( ) 1998 M. Kontsevich ( ) 21

5.3? = 1. 2. 7! 7: 1990 KdV 15 16 ( ; ) 15 V. Drinfeld 1985 16 22

( ) : http://www.rkmath.rikkyo.ac.jp/~kakei/books.html [To2], [To3] ( ) 2 3 [SN], [MJD], [Ta] [SU] [U] [ ],,, :, 13, 14, (1999);,, (2007). [ ], (1985). [ ] [Ha1] :,, 4, (,, ) 19-56, (1982). : 17, 51, 806-809 (1996). [Ha2] :, (2004). [Hi] :, (1992). [MJD],, :, [ 4] (1993); ( ) (2007). [Sh] [SN] [SU] :, 1983 8,., :, 18 (1984).,, 5, (1989). [Ta] :, (2001) [TD], : KdV, 16 (1979) 17 http://wwwsoc.nii.ac.jp/jps/jps/butsuri/50th/noframe/50(3)/50th-p806.html 23

[To1] : 18 51, 185 188, (1996) [To2] : 30, 30 3, (1999). [To3] : ( ), (2000). [U] :, 4 (1993). [W] :, 14 (1992) 18 http://wwwsoc.nii.ac.jp/jps/jps/butsuri/50th/noframe/50(3)/50th-p185.html 24