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

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1 63 KdV KP Lax pair L, B L L L / W LW / x W t, t, t 3, ψ t n / B nψ KdV B n L n/ KP B n L n KdV KP Lax W Lax τ KP L ψ τ τ Chapter 7 An Introduction to the Sato Theory Masayui OIKAWA, Faculty of Engneering, Fuuoa Institute of Technology Lax N- N- τ τ τ, 3 Lax L KdV KP oiawa@fitacjp KP Grassmann KP τ Grassmann Plücer KP Plücer KP τ τ 4 Ohta Ohta 5 5

2 64 KdV KdV 4u t uu x + u xxx 7 Lax 9 L + ux, t B 3 + 3u + 3 u x 7 / x L ψ Lψ λ ψ 73 t Bψ λ L t [B, L] BL LB 74 KdV 7 Lax KdV Lax Lax B KdV B, B u + 3 u x, B u u x u xx + 5 u u xxx + 5 uu x 74 B B i u t u x u t 4 u xx + 3 u x u t 6 u xxxx uu xx u x + 5 u3 x B i B B L, B 4 L, [B, L], [B 4, L] B L /, B 3 L 3/, B 5 L 5/ P a j x n j, a x 77 j n microdifferenial operator [B, L] n n af n r r n r a r f n r, n : nn n r + r! 78 a r ax x r x a, a ax n n a a r n r 79 r r n r n 77 n 79 n 78 n r a a a + a 3 + a a a 3 + 3a P a j x n j, Q b x m 7 P Q c i j a j x n j j j b x m a j x n j b x m j r n j a j r b r m+n j r c i m+n i 7 i j++ri n j r a j b r 73 3 c i c a b, c a b + a b + na b, c a b + a b + a b + na b +n a b nn + a b 74

3 65 L / X + f n n 75 n X + f n n + f m m n n m + f n n + f n n + m,n,l n l n f n l f m m n l 76 + u f, f, L / + u u x uxx + + u xxx uu x 4 u B n L n/ [L n/, L] B n : L n/ + L n/ 78 L n/ L n/ + B c n B n L n/ + B c n [B n, L] [L n/, L] + [B c n, L] [B c n, L] [ α n +β n +, +u] 79 α n, β n, u α n L n/ 7 74 [B n, L] + B n, L [B n, L] [B n, L] 79 [B n, L] α n x Lax L t [B n, L] u t α n x 78 B n 75 u x t, t 3, t 5, Lψ λ ψ 7 B n ψ, n, 3, 5, λ L [B n, L], n, 3, 5, 7 KdV KdV hierarchy u t u t3 u t5 u x 4 u xx + 3 u x 6 u xxxx uu xx u x + 5 u3 x 7 B n ψ B m ψ t m B n ψ B m ψ t m B n ψ B m ψ t m B n ψ B m ψ + B n B m t m t m Bn B m + [B n, B m ] ψ t m ψ B n t m B m + [B n, B m ] 73 Zaharov-Shabat Lax 3 KP L u n n 74 n u, u L + u + u L λ Lψ λψ B n ψ, B n : L n + 76

4 66 u, u 3, x t, t, t 3, Lax L [B n, L], n,, 3, 77 B n B B + u B u + 3u 3 + 3u x B u + 4u 3 + 6u x + 4u 4 + 6u 3x + 4u xx + 6u Lax 77 n 77 L [, L] L t x u it 78 u ix, i, 3, i, 3, u i x + t, t, t 3, n 77 u t + u 3 t + u 4 t u + u + u u + u u u xx + u 3x + u 3xx + u 4x + u u x +u 4xx + u 5x u u xx + 4u x u j j,, u t u xx + u 3x u 3 t u 3xx + u 4x + u u x u 4 t u 4xx + u 5x u u xx + 4u x u 3 n 3 77 u t 3 u xxx +3u 3xx +3u 4x +6u u x u 3 t 3 u 3xxx +3u 4xx +3u 5x +6u u 3x +6u x u 3 u 4 t 3 u 4xxx +3u 5xx +3u 6x 3u u 3xx 3u xx u 3 +3u u 4x +9u x u 4 +6u 3 u 3x u, u 3, u 4, u 3, u 4 u 4 u u + u x t 3 x + 3 u x u t 73 Petviashvili KP KP KP hierarchy Zaharov-Shabat 73 Lax 77 L L L + L L [B n, L]L + L[B n, L] B n L LB n L + LB n L LB n B n L L B n [B n, L ] L 3 L L + L L L + L L [B n, L]L + L[B n, L]L + L [B n, L] B n L LB n L +LB n L LB n L+L B n L LB n B n L 3 L 3 B n [B n, L 3 ] L m [B n, L m ] 73 L m Ln t m [B n, L m ] [B m, L n ] 733 L n B n Bn c Bc n L m Ln [L n +B t n, c L m ] [B m, B n Bn] c m [Bn, c B m Bm] c [B m, B n Bn] c [B n, B m ] [B c n, B c m] m, n 734 t y, t 3 t Kadomtsev- Zaharov- Shabat B m B n [B n, B m ] m, n 735 t m m, n 3 KP Bm c Bc n + [B t m, c Bn] c m, n 736 m

5 67 77 KP 77 KP Lax Lax Zaharov-Shabat 735 KP Zaharov-Shabat 4 KP W L W LW 737 W + w + w + w LW W W + w + w + w 3 + w LW + w + w + w x + u +w 3 + w x + u w + u 3 + w 4 + w 3x +u w u w x + u 3 w + u j j,, w x + u w x + u w + u w 3x + u w u w x + u 3 w + u 4 u, u 3, w, w, x C + c + c + 74 W W C C W C W C W C C W W W L Lax 77 W L W W L W W W W W W [ W ] W, L 74 [ W ] W B n, L n 74 B n L n + Bn c [ W ] W Bn c, L n 743 W 74 W W B t n, c Bn c W n W n 744 n B c n B n L n, L n W n W 74 W B n W W n, B n W n W + n 745 L W W W x C W W C W W B c nw, n 746 L, B n, B c n W W C 746 W C + W C B t nw c C, n 747 n C C W B c W t nw W C, n 748 n C F n W BnW c W W, n 749 F n 743 [ W ] W W W B c t nw, W n [, F n ] F n / x, n F n x 748 C F n C n 75

6 68 F n, C t, t, t 3, C/ t m C/ t m 75 F m F n t m + [F m, F n ] m, n Bn c Zaharov-Shabat 736 W Bn c W W W W B t nw c n W + W W B t nw c F n n [ ] F n, F m t m W [ B c n, ] Bm c W 75 t m Lax 77 W 745 KP W w, w, 738 W m + w + w + + w m m W m m fx m +w m ++w m fx 754 m f x, f x,, f m x, j,,, m f j xξ j +! ξj x+! ξj x ran m m ξ ξ ξ m Ξ ξ ξ ξ m 756 W m m x! x! Ξ 757 R m Ξ ΞR 757 Ξ {ran m m }/GLm, C 758 GLm, C m Grassmann GMm, Λ expxλ I + xλ + x! Λ + x x /! x 3 /3! x x /! x Hx : expxλξ f f f m f f f m f f f m w, w, w m m f w + m f w ++f w m m f m f m w + m f m w ++f m w m m f m w j j,, m f,, f m

7 69 m f m f f m f m m f m f m w j m f m j f f m f m m j f m f m W m f f m m m f m f m m f m f m W m f f m m f m f m j 76, 763 Hx m Wronsian w j j,, t t, t, 754 f j t, t, f j f j x; t f j x; t, t, 76 H Hx; t Hx; t Hx; t expxλ exp ηt, Λ Ξ 764 ηt, Λ t n Λ n 765 n Λ exp{x+t Λ+t Λ +t 3 Λ 3 + } p n x+t, t, t 3, p n Λ n 766 n ν +ν +ν +3ν 3 + n ν,ν,ν,ν 3, x ν t ν tν ν!ν!ν! 767 p n p p x + t p x + t + t p 3 6 x + t 3 + x + t t + t p n t m p n m, m < p n 769 p n x p n 77 p n 5 Hx; t p p p 3 ξ ξ ξ m p p Hx; t p ξ ξ ξ m ξ ξ ξ m h x; t h x; t hm x; t h x; t h x; t hm x; t 77 h j n x; t h j x; f j x 77 h j n x; t hj x; t n h j x; t x n 773 h j x; t hx; f j x 774 n x n hx; t, n,, W m x; t m h j x; t m + w x; t m + + w m x; th j x; t, j,,, m 776

8 7 w j x; t W m x; t h m h m h h m m h m m h m w j x; t h m h m j h h m m h m m j h m W m x; t h h m h m h m m h m h m m m h h m h m h m m Ohta 5 777, 778 τ w j τ t x t x 6 Lax Lax Lψ λψ B n ψ, B n L n + λ ψ L W W 779 λψ 78 ψ W ψ ψ gt, t 3, e λx 78 ψ + w λ + w λ + gt, t 3, e λx 783 g g λ B n L n + Bn c 779 L n + B c nψ 784 Bn c j,, L j j ju j j ṽ jl L l 786 lj ṽ jl t, t, 784 L n + v n L + v n L + ψ 787 v nj t, t, 779 L j ψ λ j ψ 787 λ n + v n λ + v n λ + ψ 788 log ψ λ n + v n λ + v n λ log ψ λ j t j + t + v j λ j, t : 79 j j log ψ λ 79 v j t, t, 79 ψ exp v j λ j expt + λt + λ t + 79 j exp j v jλ j /λ t t ψ + w λ + w λ + expt +λt +λ t + 79

9 7 779 w j KP KdV Boussinesq Lax l L l L l B l 793 L l B l,, B l ψ λ l ψ,, t l λ l ψ,, w j t l j,,, 797 w j t l, t l, t 3l, l Schrödinger + u ψ λ ψ L B L u 3 u x u 4 4 u xx u u 5 8 u xxx + 3 u u x 799 u u L / L 779 n u + 3 t 3 u x ψ 7 n 5 t 5 B 5 ψ 7 KdV 3-3 B 3 ψ 3 + 3u + 3u 3 + 3u x ψ λ 3 ψ 7 ψ t B ψ + u ψ 73 Boussinesq u t 4 u 3 x 4 u x 74 7 τ 5 τx; t h h m h h m h m h m m 75 τ w j τ τ Hx; t m τt det p p p 3 p p ξ ξ ξ m ξ ξ ξ m ξ ξ ξ m detξ t exp ηt, ΛΞ 76 Ξ t m Ξ t 77

10 7 t x x 76 p n p n t ν +ν +3ν 3 + n ν,ν,ν 3, t ν tν tν 3 3 ν!ν!ν 3! 78 Binet-Cauchy τt l <l < <l m ξ l ξ l ξ l ξ l ξ l m ξ l m p l p l p lm p l p l p lm p l m+ p l m+ p lm m+ ξ m l ξ m l 79 ξ m l m m n < p n l, l,, l m i m ii l m +, l m +,, l m m +, 3, 5, 6, 3, 5, 6 m m 3 3 m , 3, 5, 6, 3, 5, 6 4 φ m 79 p j Schur p l p l p lm p l p l p lm S Y t, 7 p l m+ p l m+ p lm m+ ξ Y ξ l ξ l ξ l ξ l ξ l m ξ l m ξ m l ξ m l ξ m l m 7 Y l, l,, l m 79 τt rowy m S Y tξ Y 7

11 73 m m τtdet p p p p p p p ξ ξ p ξ ξ + p p p p p 3 ξ ξ + p ξ 3 ξ + 3 ξ ξ ξ ξ ξ ξ ξ ξ S φ ξ φ +S ξ +S ξ + 73 ft S Y t ft Y S Y tc Y 74 c Y c Y S Y t ft t 75 S Y t S Y t t Ohta 5 t 7 τt S Y t 74 7 τ ξ Y Plücer m τ 73 4, l < l < ξ ξ l ξ l ξ ξ ξ l ξ l ξ ξ l ξ l ξ ξ l ξ l ξ ξ ξ ξ l ξ l ξ ξ l ξ l ξ 76 3 ξ ξ l ξ ξ ξ l ξ l ξ l ξ ξ ξ l ξ ξ ξ l ξ l ξ l ξ ξ ξ l + 3 ξ ξ ξ l ξ l ξ l ξ 77 l ξ Y, l, l,,,, 3 77 ξ φ ξ ξ ξ + ξ ξ 78, l, l,,,, 4 ξ φ ξ ξ ξ + ξ ξ, 79, l, l,,, 3, 4 ξ φ ξ ξ ξ + ξ ξ, 7, l, l,,, 3, 4 ξ ξ ξ ξ + ξ ξ, 7, l, l,,, 3, 4 ξ ξ ξ ξ + ξ ξ, 7 m < < < m l < l < < l m+ ξ ξ ξ m ξ ξ m l ξ ξ m ξ m m ξ ξ m l ξ m l l ξ ξ m l l ξ l m+ ξ m l m+ l ξ l m+ ξ m l m+ ξ m l 73 m+ δ ξ Y ξ Y 74 i δ m + i j 75 Y,, j, l i, j+,, m Y l,, l i, l i+,, l m+ j < l i < j+ ν l ν 74 ξ Y Plücer ft ft Y S Y ξ Y ξ Y Plücer ft τ 4 ft 76 7 ξ Y τ Plücer τt 76 τt + s detξ t e ηt,λ e ηs,λ Ξ 76

12 74 Ξs : e ηs,λ Ξ 77 w j τ τt + s Y S Y tξ Y s 77 w j τ p j t τ 733 ξ Y s 7 s s, s, s 3, Plücer 77 S Y t S Y Ohta 5 S Y t S Y t t δ Y Y ξ Y s S Y t τt + s t S Y s τt + s t S Y s τs δ {S Y t τt}{s Y t τt} 79 δ τ 78 S φ t τts t τt S t τts t τt + S t τts t τt 73 S φ, S t, S t / + t, S t / t, S t 3 t 3, S t 4 / t t 3 + t, 3,,,,, 3, m l, l 3 p p 3 S p p p p p 3 t 4 / t t 3 + t τ 4 t t t 3 t τ 3 τ t 3 t 3 τ t 3 { + + }{ τ t } τ 73 t t t 6 4D t3 D t +3D t +D 4 t τt τt 73 t x KP τ KP 5 p j t p j t t t w τ τ x w τ τ x τ t w 3 3 τ 6τ x 3 3 τ + τ x t t Lax 739 τ u x log τ 3 u 3 x 3 + log τ x t u x x + log τ t x t 3 x log τ KP τ { τ + p t τ λ ψ τ { τ ν ν! t λ e t +λt +λ t + { exp τ + p t τ λ + ν ν! ν λ t λ t τ t λ, t λ, ψ τt, t, } e t +λt +λ t + t ν } λ τ } τ e t +λt +λ t + e t+λt+λ t τ

13 75 ft 74 Schur ft KP τ Plücer 4 KP Grassmann 6 Ξt e ηt,λ Ξ 738 KP Grassmann e ηt,λ KP {expηt, Λ} Grassmann Zaharov, V E & Shabat, A B: A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problemi, Funct Anal Appl Kadomtsev, B B & Petviashvili, V I: On the stability of solitary waves in wealy dispersing media, Sov Phys Dolady : I : : 96 4 Sato, M & Sato, Y: Soliton equations as dynamical systems on infinite dimensional Grassmann manifold, Nonlinear Partial Differential Equations in Applied Science: proceedings of the US-Japan seminar, ed Fujita, H, Lax, P D & Strang, G Kinouniya/North- Holland, Toyo, :, : 99 Sato, M: Soliton equation as dynamical systems on a infinite dimensional Grassmann manifolds, RIMS Koyurou Kyoto University : No8 984, 4 :, No Ohta, Y, Satsuma, J, Taahashi, D & Toihiro, T: An elementary introduction to Sato theory, Progr Theor Phys Suppl No : No : KP UC No ,, :, 993, 7 9 Lax, P D: Integrals of nonlinear equations of evolution and solitary waves, Comm Pure Appl Math

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