Date Wed, 20 Jun (JST) From Kuroki Gen Message-Id Subject Part 4

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1 Part generator 3 3 L 7 4 Manin triple 8 5 KP Hamiltonian 10 6 n-component KP 12 7 nonlinear Schrödinger Hamiltonian 13 http//wwwmathtohokuacjp/ kuroki/hyogen/soliton-4txt TEX , 1

2 Date Wed, 20 Jun (JST) From Kuroki Gen Message-Id Subject Part 4 http//wwwmathtohokuacjp/~kuroki/hyogen/soliton-10txt http//wwwmathtohokuacjp/~kuroki/hyogen/soliton-11txt http//wwwmathtohokuacjp/~kuroki/hyogen/soliton-2txt http//wwwmathtohokuacjp/~kuroki/hyogen/soliton-3txt Part 3 Lie algebra Hamiltonian KP nonlinear Schrödinger Lie algebra Hamiltonian, R C Nonlinear Schrödinger (NLS) Hamiltonian 7 NLS Davey-Stewartson (DS) ( 2-component KP) Hamiltonian 1 R derivation = / x (differential algebra) (R, ), R associative algebra, R R (fg) = (f)g + f (g) (f, g R), f = (f), f (n) = n (f), etc R (subalgebra C of constants) C = { f R f = (f) = 0 } R, C 11, (1) R = R 1 = C[[x]], C = C (2) R = R 2 = M(n, R 1 ), C = M(n, C) (3) R = R 3 = C((x)), C = C (4) R = R 4 = M(n, R 3 ) = M(n, C((x))), C = M(n, C) (5) R = R 5 = C (S 1 ) = {S 1 C }, C = C (6) R = R 6 = M(n, R 5 ) = M(n, C (S 1 )), C = M(n, C) 2

3 (R, ) D = D R E = E R R D = D R = R[ ] = { M a i i M Z 0, a i R }, m=0 E = E R = R(( 1 )) = { a i i M Z, ai R } m m D E f = f + f, 1 f = f 1 f 2 + f 3 f 4 +, f R, D E associative algebra with 1 E 1 -adic topology, x n f = ( ) n f (k) x n k, f x n = ( ) n ( 1) k f (k) x n k k k k 0 k 0, ( n k) E Lie subalgebras E +, E E + = D, E = R[[ 1 ]] 1, E = E + E ( ) 2 generator 21 n, n { A E n A = A n } = C(( 1 )) A E A = i M A i i, A i R (A i = 0 if i > M) n A = A n = A i C (i M) ), Leibnitz, [ n, A] = i M ( n j [ n, A i ] i = i M n 1 ( ) n A (n j) i i+j = j j=0 k M+n 1 n 1 j=0 ( ) n A (n j) k j k j 3

4 , i > M A i = 0, [ n, A] = 0, ( ) n 0 = A n 1 M A M C 0 = ( ) ( ) ( ) n n n A M + A M 1 = A n 2 n 1 n 1 M 1 A M 1 C 0 = ( ) ( ) ( ) n n n A M + + A M 2 = A n 3 n 1 n 1 M 2 A M 2 C, i M A i C 22 a, b m, n a m b n, A E, (a m )A = A(b n ) = A = 0 A E A = i M A i i, A i R (A i = 0 if i > M) A 0 A M 0, (a m )A = a i M A(b n ) = b k M+n m j=0 ( ) m A (m j) i i+j = a j k M+m A k n k m j=0 ( ) m A (m j) k j k, j a m b n (1) a 0 b = 0 (2) a = 0 b 0 (3) ab 0 m > n (4) ab 0 m < n (5) ab 0 m = n a b 4

5 , (1) (2), (a m )A A(b n ) 0, (a m )A A(b n ), (3), (a m )A A(b n ) M + m, aa M 0, (a m )A A(b n ), (4), (a m )A A(b n ) M + n, ba M 0, (a m )A A(b n ), (5), (a m )A A(b n ) M + m, (a b)a M 0, (a m )A A(b n ) e 1,, e S C = { A R A = 0 } (1) e e S = 1, (2) e a e b = e a δ a,b C C e 23 C e = { A C e a A = Ae a } C e = { S a=1 e a Ae a A C } A = b,c e bae c, A C e, b c e b Ae c = 0 e a A = Ae a, 0 = e b (e a A Ae a )e c = (δ a,b δ a,c )e b Ae c, b c a = b e b Ae c = 0 c 1,, c S n 1,, n S a b = c a n1 c b n S, P D P = c 1 e 1 n c S e S n S generator P P 24, { A E P A = AP } = C e (( 1 )), A P A C e 5

6 { A E P A = AP } C e (( 1 )) C e, A E,,, P A = AP, A a,b = e a Ae b e a P Ae b = (c a n a )A a,b, e a AP e b = A a,b (c b n b ) (c a n a )A a,b = A a,b (c b n b ) 21 A a,a C(( 1 )), 22, a b A a,b = 0, 23, A C e (( 1 )) 25 F, F C F R = M(n, F ) {1,, n} {K 1,, K S }, e a = (E ij ) i K a E ii,, C = M(n, C F ) C e C e = S a=1 i,j K a C F E ij, C e K a 26 (nonlinear Schrödinger (NLS) ) 25 n = 2,,, S = 2, e 1 = E 11, e 2 = E 22 (E ij ) C e = C F e 1 C F e 2 = diag(c F, C F ) P = e 1 m e 2 m = diag( m, m ), A P, A diag(c F, C F ), S = 1, e 1 = E 11 + E 22 = ( ) 6

7 , C e = M(2, C F ), P = e 1 m = diag( m, m ), A P, A M(2, C F ) 3 L G G = 1 + E G A E, A n n, A + A 2 + A 3 +, A E, (1 A) 1 = 1 + A + A 2 + A E, G G Lie algebra E P D (1) e e S = 1 (2) e a e b = e a δ a,b (3) c 1,, c S (4) n 1,, n S (5) a b = c a na c b n b (6) P = c 1 e 1 n c S e S n S G G (C e ) G (C e ) = G C e (( 1 )) = 1 + C e [[ 1 ]] 1 W G, L = W P W 1 L, W L 31 W, V G, W P W 1 = V P V 1 = W 1 V 1 + C e [[ 1 ]] 1 W QW 1 = V QV 1 QW 1 V = W 1 V Q, 24, W 1 V G (C e ) 7

8 32 {P L } = { W P W 1 W G } = G /G (C e ) 33 (NLS ) 25 n = 2,,, S = 2, e 1 = E 11, e 2 = E 22 (E ij ) C e = C F e 1 C F e 2 = diag(c F, C F ) P = e 1 m e 2 m = diag( m, m ), W G P, W 1 + diag(c F [[ 1 ]] 1, C F [[ 1 ]] 1 ) m, S = 1, e 1 = E 11 + E 22 = ( ), C e = M(2, C F ), P = e 1 m = diag( m, m ), W G P, W 1 + M(2, C F [[ 1 ]] 1 ) m 4 Manin triple R restr (1) restr(ab) = restr(ba) (A, B R) (2) A R, restr(ab) = 0 B = 0 8

9 (3) restr(a ) = 0 (A R) 41 (1) R = C((x)), restr(f) = Res x=0 (f dx) (f R) (2) R = M(n, C((x))), restr(f ) = Res(tr(F ) dx) (F R) x=0 tr(f ) trace (3) R = C (S 1 ), restr(f) = f dx (f R) S 1 (4) R = M(n, C (S 1 )), restr(f ) = tr(f ) dx S 1 (F R) tr(f ) trace (5) R = C[[x]] (R) = R, restr R restr E trace trace(p ) = restr(a 1 ) (P = a i i E), E bilinear form, P, Q = trace(p Q) (P, Q E) 42 E trace, (1) trace(p Q) = trace(qp ) (P, Q E), (2) P E, trace(p Q) = 0 Q = 0 (3), E nondegenerate symmetric bilinear form (4), E +, E isotropic (, E +, E + = 0, E, E = 0 E + E, ) (5) AB, C = A, BC (A, B, C R) (6) [A, B], C = A, [B, C] (A, B, C R) (3), (4), (6), (E, E +, E ) Manin triple Manin triple http//wwwmathtohokuacjp/~kuroki/hyogen/classical-r-3txt r Part 3 10 Manin triples and the double of Lie bialgebras, E, E +, E Lie bialgebras 9

10 5 KP Hamiltonian, KP linear (= first) Poisson bracket Hamiltonian (quadratic Poisson bracket ), E = C((x))(( 1 )),, E trace invariant nondegenerate bilinear form, trace(a) = Res x=0 (a 1 dx) (A = a i i E), A, B = trace(ab) (A, B E) A E, A A + E + = D, A A E A = A + A, A + E + = D, A E E r-bracket [A, B] r = [A +, B + ] [A, B ] (A, B E) r-bracket E Lie algebra Lie algera E r A E A +, A r +, r E E 51 A, L, M E,, [M, L] = 0, L, [A, M] r = r +[M, L] + [r (M), L], A L, [A, M] r = [M, L], A = [M +, L], A [A, M] r = [r + (A), M] + [A, r (M)],, [M, L] = 0,, M = M + M L, [A, M] r = L, [r + (A), M] + [A, r (M)] = [M, L], r + (A) + [r (M), L], A = r +[M, L], A + [r (M), L], A = r +[M, L] + [r (M), L], A L, [A, M] r = [M, L], A [M, L] = [M + M, L] = [M +, L] 10

11 52,, E = E, E r Lie algebra [, ] r E Poisson {, } 1 {F, G} 1 (L) = L, [ F (L), G(L)] r (F, G E ), F (L) E F (L), A = df (L)(A) = [df (L + sa)/ds] s=0 {, } 1 KP linear Poisson bracket first Poisson bracket 53 (E, E +, E ) Manin triple, linear Poisson bracket, quadratic Poisson bracket Sklyanin bracket 54 E linear function a(l) = A, L = trace(al), da(l) = trace(a dl), da(l)(b) = trace(ab) = A, B, a(l) = A 55 E H m, H m (L) = 1 m + 1 trace(lm+1 ), dh m (L) = trace(l m dl), dh m (L)(A) = trace(l m A) = L m, A H m (L) = L m 51 54, 55, A E, {a, H m } 1 (L) = [(L m ), L], A = [(L m ) +, L], A, L {, H m } 1 E {L, H m } 1 (L), {L, H m } 1 (L) = [(L m ), L] = [(L m ) +, L], Hamiltonian H m Poisson bracket {, } 1 Hamilton Lax L t m = [(L m ), L] = [(L m ) +, L], KP hierarchy 11

12 6 n-component KP, E = E n = M(n, E 1 ), E 1 = C((x))(( 1 )),, E trace invariant nondegenerate bilinear form, trace(a) = Res x=0 (tr(a 1) dx) (A = A i i E), A, B = trace(ab) (A, B E) A E, A A + E + = D, A A E A = A + A, A + E + = D, A E E r-bracket [A, B] r = [A +, B + ] [A, B ] (A, B E) r-bracket E Lie algebra Lie algera E r A E A +, A r +, r E E 51 61,, E = E, E r Lie algebra [, ] r E Poisson {, } 1 {F, G} 1 (L) = L, [ F (L), G(L)] r (F, G E ), F (L) E F (L), A = df (L)(A) = [df (L + sa)/ds] s=0 {, } 1 n-component KP linear Poisson bracket first Poisson bracket 62 (E, E +, E ) Manin triple, linear Poisson bracket, quadratic Poisson bracket Sklyanin bracket n-component KP generators P P basis P = diag(c[ ],, C[ ]) D P i,m = E ii m (E ij, i = 1,, n, m = 1, 2, 3, ) 12

13 P basis t i,m, G = 1 + M(n, C((x))[[ 1 ]] 1 ) flow W t i,m = (L i,m ) W, L i,m = W P i,m W 1 (W G ) n-compinent KP hierarchy (n-ckp), A P, L A = W AW 1, L A Lax L A t i,m = [(L i,m ), L A ] = [(L i,m ) +, L A ] Q P, n-ckp Q-reduction constraint L Q (L Q ) = 0, L Q = (L Q ) + (Q-reduction) Q = c i,m P i,m P, [ ] ci,m W = 0 (Q-reduction) t i,m 7 nonlinear Schrödinger Hamiltonian, nonlinear Schrödinger (NLS) linear (= first) Poisson bracket Hamiltonian, nonlinear Schrödinger (NLS) hierarchy 2-component KP hierarchy Q = diag(, ) Q-reduction constraint Q-reduction constraint W diag(, )W 1 = diag(, ) 13

14 , 33,, W 1 + M(n, C[[ 1 ]] 1 ), W x = 0 Q-reduction, P 1,m = E 11 m = diag( m, 0), P 2,m = E 22 m = diag(0, m ), t 1,m, t 2,m derivation 1,m, 2,m 1,m = Q-reduction,, 2,m = t 1,m t 2,m ( 1,m + 2,m )W = 0,, s m = t 1,m + t 2,m, t m = t 1,m t 2,m, Q-reduction, sm = 1,m + 2,m, tm = 1,m 2,m sm W = 0, t m tm W = (L m ) W, L m = W diag( m, m )W 1 L m, Q = diag(, ), L m = Q m 1 L 1 ( ) 33, W L m C 1 + diag(c F [[ 1 ]] 1, C F [[ 1 ]] 1 ) 14

15 L m W m,, L 1 (0) = W (0) diag(, )W (0) 1, Lax L 1 well-defined tm L 1 = [(L m ), L 1 ] = [(L m ) +, L 1 ] Lax W, W (t) L 1 (t) C 71 ( ) (1-component) KP hierarchy 1 NLS hierarchy E H m H m (L) = 1 2 trace(qm 1 L 2 ) (L E), Q = diag(, ), H m linear (= first) Poisson bracket Hamilton NLS Lax, NLS L 1 L Q = diag(, ),, dh m (L) = trace(q m 1 L dl), dh m (L)(A) = trace(q m 1 LA) = Q m 1 L, A H m (L) = Q m 1 L, KP, Q = diag(, ) L, {L, H m } 1 (L) = [(Q m 1 L), L] = [(Q m 1 L) +, L], Hamilton, Q = diag(, ) L, tm L = [(Q m 1 L), L] = [(Q m 1 L) +, L], ( ) L m = Q m 1 L 1, NLS L 1 Lax 72 Q = diag(, ) L, L H m (L) = 0, dh m (L) 1 L m L = L 1 L KP L m = L m, NLS L m = Q m 1 L 15

16 73 Q diag(, ), L Q, Lax Hamiltonian t L = [(Q m L n ), L] = [(Q m L n ) +, L] H(L) = 1 n + 1 trace(qm L n+1 ) Hamilton, Lax pair (L, M + ) M Q 1,, Q K L M = f(q 1,, Q K ; L), Lax t L = [M, L] = [M +, L] Hamiltonian H(L) = trace(f (Q 1,, Q K ; L)), F (x 1,, x K ; y) = f(x 1,, x K ; y) dy Hamilton, 74 Hamiltonian Lax L Hamiltonian vector field, m, NLS L m L m = L m 1, L m Hamiltonian, H m (L) = 1 m + 1 trace(lm+1 1 ), NLS L, L L 1, L 0 = W diag(1, 1)W 1 ( ) L, L m = Q m L 0, Hamiltonian, H m (L) = 1 2 trace(qm L 2 ) 16

1 n A a 11 a 1n A =.. a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = 0 ( x 0 ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 1.1 Th

1 n A a 11 a 1n A =.. a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = 0 ( x 0 ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 1.1 Th 1 n A a 11 a 1n A = a m1 a mn Ax = λx (1) x n λ (eigenvalue problem) x = ( x ) λ A ( ) λ Ax = λx x Ax = λx y T A = λy T x Ax = λx cx ( 1) 11 Th9-1 Ax = λx λe n A = λ a 11 a 12 a 1n a 21 λ a 22 a n1 a n2