- Email: [email protected]
Hamiltonian and recursion operators for two-dimensional scalar fields
Physics Letters A 170 (1992) 405—408 North-Holland
PHYSICS LETTERS A
Hamiltonian and recursion operators for two-dimensional scalar fields A.G. Meshkov Kalmyk State University, Elista 358000, Russian Federation Received 30 March 1992; revisedmanuscript received 31 July 1992; accepted for publication 16 September 1992 Communicated by A.P. Fordy
The equation for the Noether operator is obtained. It gives the necessary conditions for complete integrability of the field equations. For several double-component models the Hamiltonian pairs and the recursion operators are presented.
We consider here the chiral-type scalar fields in
two-dimensional space—time with the following Lagrangian,
where
J’(u)[ço]y= (~_J(u+up)Y)~
L=~gap(u)u~uq+f(u).
Here u’~are the scalar fields, a = 1,
...,
m, ~
=
öua/8x
1, x1 are the light-cone variables. The functionsfand gap are assumed to be differentiable, and det (gap) 0. The summation rule over the repeated indices is also assumed. One can represent the Euler— Lagrange equations in the following form,
(1)
Japuq =fa,
wherefa=8f/ôu~8af~and
then the Jacobi identity can be verified by the direct calculation for this bracket. It means that J is a symplectic operator (see ref. [1]). Notice that a more general symplectic operator was presented in ref. [21. Asfa=8f/8u°~ is the variational derivative, and 1’ is the implectic (Hamiltonian) operator (see ref. [1]), system (1) is equivalent to an explicitly Hamiltonian one (3)
ua_(J_l)aP(sf/&us)
(2)
where ü=u2. According to refs. [1] and [3] oper-
Here and below D. is the total derivative operator with respect to x, .r’vm. =gaaI’~pare the Christoffel symbols for the metric gap. Let us introduce the standard bilinear form
ator J maps the set of symmetries E of system (3) into the set of conserved covanants Therefore J is the inverse Noether operator. Any map from F into Z is called the Noether operator. If system (1) admits a Noether operator øJ—’, then there exists a nontrivial operator A = ØJ: I-~ which is called the recursion operator. If the Noether operator 0 is im-
+IvpaU~’.
Jap=gapDi
<~,9’> =
$
b ~,a (X)9’a
(x) dx,
a
where yi and ~ are any smooth functions vanishing at the points a and b. Then one can easily verify that operator (2) is skew-symmetric: <~t’, JD> = <.J~,9’>. If we calculate now the following bracket, —
[~v,q’, y] = <~J’(u) [9’])’>,
r.
plectic and if 0 and J’ are compatible (Hamiltonian pair), then the recursion operator A = OJ is hereditary [1]. The hereditary operator generates an Abelian subalgebra {A’u1, neZ} of ALie—Bäcklund fields We [4]. shall consider now the Noether operator for system (3), but we do not require that it isimplectic.
0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
405
Volume 170, number 6
PHYSICS LETTERS A
The equation for the Noether operator 6 of the evolution system (3) is [1] (D2—K’)6_—6(D2+K’
(4)
1),
23 November 1992
find that the vectors a, and w~satisfy eq. (5), the mixed components of the tensors A, satisfy the following system, ~‘A~_2+ (J~iV+P)A1_1+A1_1R(u1, u2)
where K’ is the Frechet derivative of K=J~(öf/6u),
and K’ + is the adjoint operator. To simplify eq. (4) we write the equation for the symmetries of system (1) in the following compact form, (VW+cP)a=0,
(5)
where1~pfip a is a symmetry (Lie—Bäcklund vector field), ciiafi Da _fa;p. The semicolon denotes the covariant u~u~’ derivative on configurational space, R ~ a r’a a pa + r~r~P~I’~PVis the Rie—
—
fivp
—
p
vfl
~
j>_L~J(i+i)
(~) ~
I
+A1+1_1]
~
(:~
+i) [(1’A1+1)+A~~1_1]1”JR(u1, u2), (9)
where i ~ 1, 4~=~ aw 1 according to the definition u ~‘u Moreover the following
p
mann tensor and
and R~(u1, u2) ~
V~=ô~D1+1),u~,W~=~D2+1)~u~. (6)
equality,
Comparing the equation (D2—K’ )a=0 for the symmetries of system (3) with the equivalent equation (5), we conclude that D2 —K’ = W+ V”~1. Taking intoaccount that V~=gaVJ~p,we find V~=—gVg’, where g= (g,~),g 1 (gaP). The analogous equality for W is also valid therefore D2 +K’ + =g( w+ ~V’)g’, where Q~=g’c1.~g or more explicitly a ~a u ~ fa;p. With the help of the prefl vflp vious results one can transform eq. (4) to the followingform,
V[WA]V+c1AV_VA~=0,
(7)
where A = 6g. To solve this equation we prolong the domain of operators V and W on arbitrary tensors in the following manner, V—+ ~‘= D1 + U1~ta, W-4 g7=D2+u~r~, where fa is the connection opera1 $~t ~ afi tor. T~ Forandexample =~Yav ‘TV rv (6) TYp~ 1~ so on. In taT~p2 particular operators coincide with the prolonged ones on vector fields. It is obvious that the commutator of the operator V and —
—
~
/
~.
“
~
~ A’~Y~
\i=1
k~O
I
=0,
(10)
must be valid. The proof is obtained by a straightforward calculation with the help of the Leibniz formula for and the following commutator,
c~
[~,
~‘]
=
~
(~) [~‘
‘~
(ui, u2)] ~
(11)
where~(ui,u2)Ta=R~(ui,u2)TY,~(ui,u2)T~= R~(u1,u2)T~—R~(u1,u2)T~andsoon.Ifwehave a nontrivial Noether operator 0, then the recursion operator A = 6J commutes with D2 K’. This implies that the operators 6k=AkO, k= 1, 2, satisfy eq. (4). Hence we may assume n to be large enough in expression (8). Setting i=n+2, n+l, in eq. (9), we can find any number of consequences: WA~= 0, ~ + ~PA~ = nA~R (u 1, u2) +A~~, (12) —
...,
...,
the (1, 1)-tensor Tis [V, T]ap=PT~moreover In many examples the recursion operators are integro-differential. We assume A=6.J, where J is a
differential operator, hence we must search 6 as an integro-differential one: 2~+~r a~Dj’w~, (8) 6afl,,~ ~ A?~i fl
where A,, a, and w• depend on u’~and their derivatives. Substituting expression (8) into eq. (7), we
406
~
u2),
(13)
and so on. These equations give necessary condisolve (12), (13) it of is useful to introduce the foltions eqs. for the existence the Noether operator. To lowing variables, v~=V~v~ 1, v~=u~,
Volume 170, number 6
PHYSICS LETTERS A
23 November 1992
~
(19)
w~= W~w~1, w~’= Ua
‘~
It is obvious that we may consider A as functions of u, v1 and w1. But eqs. (12), (13), imply that 8A1/ äWk=O. If we consider the scale transformation x1—vk’x1, x2-+kx2, then the Lagrangian, the field equations (1) and (5)17—+kt2 are invariant. Under this transformation we eq. have g~’-+k’J~~ therefore v~ k”v~’.Performing the scale transformation in eq. (7), we find that the operator 0(k) satisfies ...,
—~
~
=
\
~n+ I a~a 2 a0 P~p)
)
(n+ 1 )a~R~yp,2p —
~ cv) ‘ pa 0(vy)A (20) ...flR~(~Qa~ R Here we denote for brevity G~,= P~,+ Q~, a(abp) =aabp+apba, a.nd .
a
the same equation. If we expand 6(k) in the power series 0(k) = ~ 0 1k’, then each operator 0 satisfies eq. (7). Let us consider the power expansion for some function ~(k’v1) = ~ ~, ( v) k’. Substituting v1—vA’v1 or k—~Jtkhere, we obtain two equalities with the same left hand sides. Equating them, we find
B~=G~,f~+ (n+ l)a~f ;p, ~ the semicolon denotes the covariant derivative on configurational space: f a~= Vpf a~ Notice that in eq. (20) we used the integrability condition for tensor
ço(t’k’v1)= ~ ço1(Ev1)k’= ~ ~,(v)~k)’.
Q. Moreover if we set
This implies 99,, (A’v1) = A?’ço,, ( v), hence ço,, is a homogeneous function of degree n (degco,, = n). Analogously 6,, is a homogeneous operator of degree n in the above operator expansion. As each 0,, satisfies eq. (7) we considerbelow that the Noether operator 6 isdegA,,=d, homogeneous under the scale transformation. Let then degA,,_ 1 =d+l, degA0=d+n, deg ( c~co1)= d+ n + 1. We investigate the simplest case when the A are homogeneous polynomials and d= 0:
pam, =
(~)x~, +
nx~ y~,, —
Q~, = nX~,, + X~+ Y~,,, then eqs. (16) are reduced to the following, 0 andya(17) Apy,p_ py;v—”P “ye. (21) ~a a~R —
Pvv’
—
...,
(A,,)~=a~(u), (A,,.1)~=Q~,,(u)v~, (A,,2)~=P~,(u)v~ +S~,~(u)v~v~,
(14)
and so on. To solve eqs. (12) and (13) we use the formula
l~’v,,=~’~’f— n~I(n_l)[I~ri_1R(vw)]v ~=I
(15) which follows from (11). Substituting expressions (14) intoeqs. (12) and (13) andhavinginmind eq. (15), we obtain the following equations for the tensors a, Q and P, ~
conjugateand real quantities). Then eqs. (16)— (20) have three nontrivial solutions only: ~
L
2+c)—’ 2=~(uIu~+u~u1)(u’u +ku’u2(u1u2+c),
~
Q~,A=na~R~ ~ P~ =
Equations (16)— (20) are necessary conditions for the existence of the Noether operator. But this system is overdetermined and in fact it determines g~ andf” completely. We now have the general solution of system (21) for any m and present the following example. If m = 2 we can write the metric of the configu2= rational space in the canonical form ds bitrary function (u’ and u2 may be both complex
(~)
a~R~ + na~R~
(16) —
aapR a
(17) (18)
(22) where c and k are any constants. These Lagrangians have been found in ref. [5] via the higher polynomial conservation laws. If one introduces new fields z and 2, 407
Volume 170, number 6
u’—u2=
J
PHYSICS LETTERS A
A=6J=
23 November 1992
(~ ( ?)Dl+99
2u~ _u2u{) u’u~ 2u’u~‘\ —2u
(z2 2—z2Z)dx2+(z12—z21)dx1,
+r®D~’y+w®Dr’K. 2 ( uf, u ~) and ICa = Here the)2 rows ço( çou’ (u~ u~ Ya2 = (ucço) 2_~ ~)are the gradients 1,ç0u of the conserved quantities Pi = 9’UUi and P2 = çou 2 + c; 0 is the symbol zi~,adirect ccordingly, çr~ for1 the product of=au’u column and a row. For the second model of (22) system (24) does not have a solution. But because the model is integrable we believe that the Noether operator (8) exists if n> 1. For the third model of (22) the Noether —
then other the field equations for are L3 integrable will become linear. The two models (22) by the inverse spectral transform method (see refs. [6,7]). To find the Noether operator we must choose the n in eq. (8). Let n = 0, then system (9) coincides with system (12). The first two models (22) admit the following obvious symmetries: t~X= (u ~,u~)T and = (u’, u2)T Assuming that —
6=A 0+c1tD~’w+c2wD~’r,
(23)
where c, are constants, we find the following system
from (12), cIg$~a)Aö~ c1 faWn +
A~
—
operator has the form (23), where r, A0 and c are the same, but W’= (1, 1 )T~Hence the third model has the recursion operator A = OJ. —
+c2wa;,,g~.,=aaR~—a~R~,
c2 wafp = (ap — aa )f a.~
(24) 2, then the above
[1] A.S. Fokas and B. Fuchssteiner, Lett. Nuovo Cimento 28 [2] 0.1. (1980) Mokhov, 299. Funct. Anal. Appl. 24, no. 3 (1990)86 [in
1 =c2=a2=l, a1=—l, and system (10) is satisfied identically. Hence operator (23) with the mentioned c and a is the Noetherian for the first system (22). A direct (but cumbersome) calculation yields that operator (23) is implectic and compatible with J j. Hence the first model (22) admits the hereditary recursion operator
Russian]. [3] B. Fuchssteiner and A.S. Fokas, Physica D 4 (1981) 47. [4] B. Fuchssteiner, Prog. Theor. Phys. 65 (1981) 861. [5] B.S. Getmanov, in: Theor. Group Math. Phys. Proc. Int. Sem., Zvenigorod, 24—26 November 1982, Vol. 2 (Nauka, Moscow, 1983) p. 333. [6] F. Lund, Ann. Phys. 115 (1978) 251. [7] B.S. Getmanov, Teor. Mat. Fiz. 48 (1981) 13.
=
~
References
Here aaimplies are any constants. Iff= k&u system c
408
PHYSICS LETTERS A
Hamiltonian and recursion operators for two-dimensional scalar fields A.G. Meshkov Kalmyk State University, Elista 358000, Russian Federation Received 30 March 1992; revisedmanuscript received 31 July 1992; accepted for publication 16 September 1992 Communicated by A.P. Fordy
The equation for the Noether operator is obtained. It gives the necessary conditions for complete integrability of the field equations. For several double-component models the Hamiltonian pairs and the recursion operators are presented.
We consider here the chiral-type scalar fields in
two-dimensional space—time with the following Lagrangian,
where
J’(u)[ço]y= (~_J(u+up)Y)~
L=~gap(u)u~uq+f(u).
Here u’~are the scalar fields, a = 1,
...,
m, ~
=
öua/8x
1, x1 are the light-cone variables. The functionsfand gap are assumed to be differentiable, and det (gap) 0. The summation rule over the repeated indices is also assumed. One can represent the Euler— Lagrange equations in the following form,
(1)
Japuq =fa,
wherefa=8f/ôu~8af~and
then the Jacobi identity can be verified by the direct calculation for this bracket. It means that J is a symplectic operator (see ref. [1]). Notice that a more general symplectic operator was presented in ref. [21. Asfa=8f/8u°~ is the variational derivative, and 1’ is the implectic (Hamiltonian) operator (see ref. [1]), system (1) is equivalent to an explicitly Hamiltonian one (3)
ua_(J_l)aP(sf/&us)
(2)
where ü=u2. According to refs. [1] and [3] oper-
Here and below D. is the total derivative operator with respect to x, .r’vm. =gaaI’~pare the Christoffel symbols for the metric gap. Let us introduce the standard bilinear form
ator J maps the set of symmetries E of system (3) into the set of conserved covanants Therefore J is the inverse Noether operator. Any map from F into Z is called the Noether operator. If system (1) admits a Noether operator øJ—’, then there exists a nontrivial operator A = ØJ: I-~ which is called the recursion operator. If the Noether operator 0 is im-
+IvpaU~’.
Jap=gapDi
<~,9’> =
$
b ~,a (X)9’a
(x) dx,
a
where yi and ~ are any smooth functions vanishing at the points a and b. Then one can easily verify that operator (2) is skew-symmetric: <~t’, JD> = <.J~,9’>. If we calculate now the following bracket, —
[~v,q’, y] = <~J’(u) [9’])’>,
r.
plectic and if 0 and J’ are compatible (Hamiltonian pair), then the recursion operator A = OJ is hereditary [1]. The hereditary operator generates an Abelian subalgebra {A’u1, neZ} of ALie—Bäcklund fields We [4]. shall consider now the Noether operator for system (3), but we do not require that it isimplectic.
0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
405
Volume 170, number 6
PHYSICS LETTERS A
The equation for the Noether operator 6 of the evolution system (3) is [1] (D2—K’)6_—6(D2+K’
(4)
1),
23 November 1992
find that the vectors a, and w~satisfy eq. (5), the mixed components of the tensors A, satisfy the following system, ~‘A~_2+ (J~iV+P)A1_1+A1_1R(u1, u2)
where K’ is the Frechet derivative of K=J~(öf/6u),
and K’ + is the adjoint operator. To simplify eq. (4) we write the equation for the symmetries of system (1) in the following compact form, (VW+cP)a=0,
(5)
where1~pfip a is a symmetry (Lie—Bäcklund vector field), ciiafi Da _fa;p. The semicolon denotes the covariant u~u~’ derivative on configurational space, R ~ a r’a a pa + r~r~P~I’~PVis the Rie—
—
fivp
—
p
vfl
~
j>_L~J(i+i)
(~) ~
I
+A1+1_1]
~
(:~
+i) [(1’A1+1)+A~~1_1]1”JR(u1, u2), (9)
where i ~ 1, 4~=~ aw 1 according to the definition u ~‘u Moreover the following
p
mann tensor and
and R~(u1, u2) ~
V~=ô~D1+1),u~,W~=~D2+1)~u~. (6)
equality,
Comparing the equation (D2—K’ )a=0 for the symmetries of system (3) with the equivalent equation (5), we conclude that D2 —K’ = W+ V”~1. Taking intoaccount that V~=gaVJ~p,we find V~=—gVg’, where g= (g,~),g 1 (gaP). The analogous equality for W is also valid therefore D2 +K’ + =g( w+ ~V’)g’, where Q~=g’c1.~g or more explicitly a ~a u ~ fa;p. With the help of the prefl vflp vious results one can transform eq. (4) to the followingform,
V[WA]V+c1AV_VA~=0,
(7)
where A = 6g. To solve this equation we prolong the domain of operators V and W on arbitrary tensors in the following manner, V—+ ~‘= D1 + U1~ta, W-4 g7=D2+u~r~, where fa is the connection opera1 $~t ~ afi tor. T~ Forandexample =~Yav ‘TV rv (6) TYp~ 1~ so on. In taT~p2 particular operators coincide with the prolonged ones on vector fields. It is obvious that the commutator of the operator V and —
—
~
/
~.
“
~
~ A’~Y~
\i=1
k~O
I
=0,
(10)
must be valid. The proof is obtained by a straightforward calculation with the help of the Leibniz formula for and the following commutator,
c~
[~,
~‘]
=
~
(~) [~‘
‘~
(ui, u2)] ~
(11)
where~(ui,u2)Ta=R~(ui,u2)TY,~(ui,u2)T~= R~(u1,u2)T~—R~(u1,u2)T~andsoon.Ifwehave a nontrivial Noether operator 0, then the recursion operator A = 6J commutes with D2 K’. This implies that the operators 6k=AkO, k= 1, 2, satisfy eq. (4). Hence we may assume n to be large enough in expression (8). Setting i=n+2, n+l, in eq. (9), we can find any number of consequences: WA~= 0, ~ + ~PA~ = nA~R (u 1, u2) +A~~, (12) —
...,
...,
the (1, 1)-tensor Tis [V, T]ap=PT~moreover In many examples the recursion operators are integro-differential. We assume A=6.J, where J is a
differential operator, hence we must search 6 as an integro-differential one: 2~+~r a~Dj’w~, (8) 6afl,,~ ~ A?~i fl
where A,, a, and w• depend on u’~and their derivatives. Substituting expression (8) into eq. (7), we
406
~
u2),
(13)
and so on. These equations give necessary condisolve (12), (13) it of is useful to introduce the foltions eqs. for the existence the Noether operator. To lowing variables, v~=V~v~ 1, v~=u~,
Volume 170, number 6
PHYSICS LETTERS A
23 November 1992
~
(19)
w~= W~w~1, w~’= Ua
‘~
It is obvious that we may consider A as functions of u, v1 and w1. But eqs. (12), (13), imply that 8A1/ äWk=O. If we consider the scale transformation x1—vk’x1, x2-+kx2, then the Lagrangian, the field equations (1) and (5)17—+kt2 are invariant. Under this transformation we eq. have g~’-+k’J~~ therefore v~ k”v~’.Performing the scale transformation in eq. (7), we find that the operator 0(k) satisfies ...,
—~
~
=
\
~n+ I a~a 2 a0 P~p)
)
(n+ 1 )a~R~yp,2p —
~ cv) ‘ pa 0(vy)A (20) ...flR~(~Qa~ R Here we denote for brevity G~,= P~,+ Q~, a(abp) =aabp+apba, a.nd .
a
the same equation. If we expand 6(k) in the power series 0(k) = ~ 0 1k’, then each operator 0 satisfies eq. (7). Let us consider the power expansion for some function ~(k’v1) = ~ ~, ( v) k’. Substituting v1—vA’v1 or k—~Jtkhere, we obtain two equalities with the same left hand sides. Equating them, we find
B~=G~,f~+ (n+ l)a~f ;p, ~ the semicolon denotes the covariant derivative on configurational space: f a~= Vpf a~ Notice that in eq. (20) we used the integrability condition for tensor
ço(t’k’v1)= ~ ço1(Ev1)k’= ~ ~,(v)~k)’.
Q. Moreover if we set
This implies 99,, (A’v1) = A?’ço,, ( v), hence ço,, is a homogeneous function of degree n (degco,, = n). Analogously 6,, is a homogeneous operator of degree n in the above operator expansion. As each 0,, satisfies eq. (7) we considerbelow that the Noether operator 6 isdegA,,=d, homogeneous under the scale transformation. Let then degA,,_ 1 =d+l, degA0=d+n, deg ( c~co1)= d+ n + 1. We investigate the simplest case when the A are homogeneous polynomials and d= 0:
pam, =
(~)x~, +
nx~ y~,, —
Q~, = nX~,, + X~+ Y~,,, then eqs. (16) are reduced to the following, 0 andya(17) Apy,p_ py;v—”P “ye. (21) ~a a~R —
Pvv’
—
...,
(A,,)~=a~(u), (A,,.1)~=Q~,,(u)v~, (A,,2)~=P~,(u)v~ +S~,~(u)v~v~,
(14)
and so on. To solve eqs. (12) and (13) we use the formula
l~’v,,=~’~’f— n~I(n_l)[I~ri_1R(vw)]v ~=I
(15) which follows from (11). Substituting expressions (14) intoeqs. (12) and (13) andhavinginmind eq. (15), we obtain the following equations for the tensors a, Q and P, ~
conjugateand real quantities). Then eqs. (16)— (20) have three nontrivial solutions only: ~
L
2+c)—’ 2=~(uIu~+u~u1)(u’u +ku’u2(u1u2+c),
~
Q~,A=na~R~ ~ P~ =
Equations (16)— (20) are necessary conditions for the existence of the Noether operator. But this system is overdetermined and in fact it determines g~ andf” completely. We now have the general solution of system (21) for any m and present the following example. If m = 2 we can write the metric of the configu2= rational space in the canonical form ds bitrary function (u’ and u2 may be both complex
(~)
a~R~ + na~R~
(16) —
aapR a
(17) (18)
(22) where c and k are any constants. These Lagrangians have been found in ref. [5] via the higher polynomial conservation laws. If one introduces new fields z and 2, 407
Volume 170, number 6
u’—u2=
J
PHYSICS LETTERS A
A=6J=
23 November 1992
(~ ( ?)Dl+99
2u~ _u2u{) u’u~ 2u’u~‘\ —2u
(z2 2—z2Z)dx2+(z12—z21)dx1,
+r®D~’y+w®Dr’K. 2 ( uf, u ~) and ICa = Here the)2 rows ço( çou’ (u~ u~ Ya2 = (ucço) 2_~ ~)are the gradients 1,ç0u of the conserved quantities Pi = 9’UUi and P2 = çou 2 + c; 0 is the symbol zi~,adirect ccordingly, çr~ for1 the product of=au’u column and a row. For the second model of (22) system (24) does not have a solution. But because the model is integrable we believe that the Noether operator (8) exists if n> 1. For the third model of (22) the Noether —
then other the field equations for are L3 integrable will become linear. The two models (22) by the inverse spectral transform method (see refs. [6,7]). To find the Noether operator we must choose the n in eq. (8). Let n = 0, then system (9) coincides with system (12). The first two models (22) admit the following obvious symmetries: t~X= (u ~,u~)T and = (u’, u2)T Assuming that —
6=A 0+c1tD~’w+c2wD~’r,
(23)
where c, are constants, we find the following system
from (12), cIg$~a)Aö~ c1 faWn +
A~
—
operator has the form (23), where r, A0 and c are the same, but W’= (1, 1 )T~Hence the third model has the recursion operator A = OJ. —
+c2wa;,,g~.,=aaR~—a~R~,
c2 wafp = (ap — aa )f a.~
(24) 2, then the above
[1] A.S. Fokas and B. Fuchssteiner, Lett. Nuovo Cimento 28 [2] 0.1. (1980) Mokhov, 299. Funct. Anal. Appl. 24, no. 3 (1990)86 [in
1 =c2=a2=l, a1=—l, and system (10) is satisfied identically. Hence operator (23) with the mentioned c and a is the Noetherian for the first system (22). A direct (but cumbersome) calculation yields that operator (23) is implectic and compatible with J j. Hence the first model (22) admits the hereditary recursion operator
Russian]. [3] B. Fuchssteiner and A.S. Fokas, Physica D 4 (1981) 47. [4] B. Fuchssteiner, Prog. Theor. Phys. 65 (1981) 861. [5] B.S. Getmanov, in: Theor. Group Math. Phys. Proc. Int. Sem., Zvenigorod, 24—26 November 1982, Vol. 2 (Nauka, Moscow, 1983) p. 333. [6] F. Lund, Ann. Phys. 115 (1978) 251. [7] B.S. Getmanov, Teor. Mat. Fiz. 48 (1981) 13.
=
~
References
Here aaimplies are any constants. Iff= k&u system c
408