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Inverse recursion operator of the AKNS hierarchy
PhysicsLettersA 179 (1993) 271—274 North-Holland
PHYSICS LETTERS A
Inverse recursion operator of the AKNS hierarchy Sen-yue Lou CCAST(World Laboratory), P.O. Box 8730, Beijing 100080, China and Institute of Modern Physics, Ningbo Normal College, Ningbo 315211, China
and Wei-zhong Chen Institute of Modern Physics, Ningbo Normal College, Ningbo315211, China Received21 October 1992; accepted for publication 8 June 1993 Communicated by D.D. Hoim
The inverse ofthe recursion operator for the AKNS system is obtained by virtue of Riccati-type equations. The nonisospectral AKNS hierarchy is extended to a full hierarchy while the isospectral is only a half hierarchy.
Owing to the excellent work of Olver [1], the recursion operator of an integrable model plays an important role. When a recursion operator of a model is obtained, various integrable properties such as bi-Hamiltonian structure, infinitely many symmetries and infinitely many conserved quantities etc. will follow naturally. Since the recursion operator is an integro-differential operator usually, it is difficult to get the explicit inverse of the known recursion operator and then one has to apply the inverse recursion operator only formally [2—4].However, the formal use of the inverse recursion operator will lead to some ambiguities or even errors (see below). Recently, we have obtained the explicit inverse recursion operators forthe KdV, mKdV, Caudry—Dodd— Gibbon— Sawada—Kortera models and other (1 + 1)-dimensional integrable ones [5—7]. In this Letter we would like to give the inverse recursion operator for the well known AKNS hierarchy, u~=K~ (n=O,l,2,...),
(1)
where K~=rI.~K0 (n=O,l,2,...),
(2)
~=(~)~ K~=(.”).
(3)
The recursion operator reads ~i(—D+2qD’r i \. —2rD’r —
2qD’q D—2rD’q
4 ‘
(
with D_—ô/Ox and D’=f~ dx. To begin the calculation, we set Mailing address. 0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
271
Volume 179, number 4,5
(A
~
PHYSICS LETTERS A
16 August 1993
B\
‘k~c E)’
(5)
such that the operators A, B, C and E are be determined by
~).
~_‘~=~_‘=i=(~ From
(6)
~ ~=I we have ‘
A(—D+2qD’r)—2BrD~r=l
(7)
,
2AqD’q+B(D—2rD’q)=O,
(8)
C(—D+2qD’r)—2ErD’r=0, 2CqD1q+E(D—2rD’q)=l . Acting eq. (8) with q ‘Dq —
—‘
(9) (10)
from the right yields
A= —~B(D—2rD’q)qDq’_——~B(Dq’Dq~—2rq’).
(11)
Substituting (11) into (7), we get B=2q’[Dq’Dq’Dq’—4q’rDq’
—2(q’r)~q’] 1=2[(D—a)q’Dq’(D+a)]1
=2F’D’FqD’qFD~F’,
(12)
with F=expD~a,
(13)
where a is related to r and q by the following Riccati equation, a~=~a2+q’q~a—2qr.
(14)
Combining eqs. (11) and (12), one obtains A=—FD’FqD’qFD~F’(Dq’Dq’—2rq’).
(15)
In the same way, the operators C and E read C=—2G’D’GrD’GrD~G’,
(16)
E=GD’GrD’rGD’G’(Dr’Dr’—2qr’)
(17)
,
with G=expD~fl,
(18)
while fi is also given by a Riccati equation, fl~=~fl2+r’r~fl—2qr.
(19)
Finally, the left inverse recursion operator for the AKNS hierarchy reads
.(—F 1D ‘FqD ‘qFEr ‘F—’ (Dq— ‘Dq —2G’D’GrD’rGD’G’ —
—
—
2rq ‘)
2F ‘D ‘FqD ‘qFD ‘F G’D’GrD’rGD1G’(Dr’Dr1—2qr’) 1
(20) 272
Volume 179, number 4,5
PHYSICS LETTERS A
16 August 1993
It is easy to prove that 1J given by eq. (20) is also the right inverse operator of ‘~P. At first sight, one can extend the half AKNS hierarchy (1) to a full hierarchy by taking n in (10) being negative and positive integers just as in the formal inverse recursion operator theory. However, there is no other nontrivial equation in the full AKNS hierarchy than that in the half AKNS hierarchy because ‘
s~J~’K
(21)
0=0.
The symmetries of the AKNS hierarchy (1) have been studied widely in the literature [8,9]. The known symmetries are mKo (m=0,l,2,...), (22) Km~I’
t~mI’mT~j~J’m(fltKn....t +XU)fltKn+m....i +‘lm(xu)
(m=0, 1,2,...).
(23)
Now the i~, symmetries can be extended to the negative direction with Km = 0 for m <0. This fact shows us that the nonisospectral AKNS hierarchy [8] can be extended to the full hierarchy u~=r~, (n=0, 1,2,..., m=0, ±1,±2,...),
(24)
though the isospectral AKNS hierarchy is a half one only. The Lie algebra constituted by Km and i~, for the AKNS hierarchy now reads [Km,Ki]0 (m,10, 1,2,...), (m,n=0,l,2,...,1=0,±l,±2,...),
[Km,t7]mKm+i..i
[ta,, il] = (m—1)T~,+,..,
(25) (26)
(n=0, 1,2,..., m, 1=0, ± 1, ±2,...)
,
(27)
with K~=Ofor p
D_1=Jdx,
one can get various new symmetries of models [5—7].However, we fail to get any other new symmetries for the general AKNS hierarchy (1). Finally we will take a special case of (1), say the mKdV hierarchy, r= q and q a real function of x and t. Then the nontrivial equations of (1) read U~=tP2m÷lU
(m,l=0,l,2,...),
(29)
which can be reduced the the scalar equations [8] (30) where Ro=q~,
(31)
~,=_q~+6q2q~,
and k,=_D2+4q~D—’q+4q2.
(32)
At a first glance, m = I in eq. (30) will give a nontrivial integrable model, say the sinh-Gordon equation [3,8]. Actually, this is not true in the real situation. The inverse of ~, [5] is —
‘I’ j~= —q’D(D+2q)’q(D—2q)’D’
(g=exp2D~q) .
= ~gD’g’D’ —
—
(33) 273
Volume 179, number 4,5
PHYSICS LETTERS A
A simple calculation will show that eq. (30) with m =
—
16 August 1993
1 is a trivial equation, (34)
q~=0,
rather than the sinh-Gordon equation. The known sinh-Gordon equation comes from another symmetry hierarchy of the mKdV equation (m = for eq. (30)). In ref. [7], we have given four other hierarchies of new symmetries of the mKdV equation. These symmetries can be rewritten as £~=~j-msinh(q) =~~gD’g1,
~~=~jmcosh(q) ~=~~j~’g’D’g
(m=0,l,2,...), (n=l,2,...) ,
(35) (36)
by combining them linearly. From the first hierarchy of symmetries we get the sinh-Gordon hierarchy q
(m=0, 1,2,...) . (37) The first equation of (37) is just the known sinh-Gordon equation. It is also easy to check that acting ~ on will result in a trivial symmetry instead of u~. In summary, the explicit form of the inverse recursion operator indicates how to get diverse correct new information on the integrable hierarchies. In this Letter, the explicit inverse recursion operator of the AKNS hierarchy is obtained. The nonisospectral AKNS hierarchy is extended to a full hierarchy while the isospectral one is only a half hierarchy. Some types of new infinitely many symmetries can be obtained for special cases, while how to get the new symmetry hierarchies for the general AKNS hierarchy should be studied further. 1=i.~
~~‘)
This work was supported by the Natural Science Foundation of Zhejiang Province and the National Science Foundation of China. The authors would like to thank Professors G. Ni, D. Chen and Dr. H. Ruan for discussions.
References [1]P.J.Olver,J. Math. Phys. 18(1977)1212. [2J B. Fuchssteiner, Nonlinear Analysis TMA 3 (1979) 849. [3] B. Fuchssteiner, Prog. Theor. Phys. 65 (1981) 861. [4]5. Carillo and B. Fuchssteiner, J. Math. Phys. 30 (1989)1606. [5J S. Lou, The fractorization and inverse of the recursion operators for some integrable models, to appear in: Proc. XXI Int. Conf. on Differential geometric methods in theoretical physics, Tianjing, 5—9 June 1992. [6] 5. Lou, Symmetries of KdVequation and four hierarchies ofthe integro-differential KdV equation, preprint NBN-IMP 07/92. [7] 5. Lou, Symmetries, conservation laws and infinite dimensional Lie algebra of the modified KdV equation, Preprint NI3N-IMP 10/ 92. [8]Y. Li andG. Zhu,J. Phys. A 19 (1986) 3713. [9]D.ChenandH.Zheng,J.Phys.A24(1991)377.
274
PHYSICS LETTERS A
Inverse recursion operator of the AKNS hierarchy Sen-yue Lou CCAST(World Laboratory), P.O. Box 8730, Beijing 100080, China and Institute of Modern Physics, Ningbo Normal College, Ningbo 315211, China
and Wei-zhong Chen Institute of Modern Physics, Ningbo Normal College, Ningbo315211, China Received21 October 1992; accepted for publication 8 June 1993 Communicated by D.D. Hoim
The inverse ofthe recursion operator for the AKNS system is obtained by virtue of Riccati-type equations. The nonisospectral AKNS hierarchy is extended to a full hierarchy while the isospectral is only a half hierarchy.
Owing to the excellent work of Olver [1], the recursion operator of an integrable model plays an important role. When a recursion operator of a model is obtained, various integrable properties such as bi-Hamiltonian structure, infinitely many symmetries and infinitely many conserved quantities etc. will follow naturally. Since the recursion operator is an integro-differential operator usually, it is difficult to get the explicit inverse of the known recursion operator and then one has to apply the inverse recursion operator only formally [2—4].However, the formal use of the inverse recursion operator will lead to some ambiguities or even errors (see below). Recently, we have obtained the explicit inverse recursion operators forthe KdV, mKdV, Caudry—Dodd— Gibbon— Sawada—Kortera models and other (1 + 1)-dimensional integrable ones [5—7]. In this Letter we would like to give the inverse recursion operator for the well known AKNS hierarchy, u~=K~ (n=O,l,2,...),
(1)
where K~=rI.~K0 (n=O,l,2,...),
(2)
~=(~)~ K~=(.”).
(3)
The recursion operator reads ~i(—D+2qD’r i \. —2rD’r —
2qD’q D—2rD’q
4 ‘
(
with D_—ô/Ox and D’=f~ dx. To begin the calculation, we set Mailing address. 0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
271
Volume 179, number 4,5
(A
~
PHYSICS LETTERS A
16 August 1993
B\
‘k~c E)’
(5)
such that the operators A, B, C and E are be determined by
~).
~_‘~=~_‘=i=(~ From
(6)
~ ~=I we have ‘
A(—D+2qD’r)—2BrD~r=l
(7)
,
2AqD’q+B(D—2rD’q)=O,
(8)
C(—D+2qD’r)—2ErD’r=0, 2CqD1q+E(D—2rD’q)=l . Acting eq. (8) with q ‘Dq —
—‘
(9) (10)
from the right yields
A= —~B(D—2rD’q)qDq’_——~B(Dq’Dq~—2rq’).
(11)
Substituting (11) into (7), we get B=2q’[Dq’Dq’Dq’—4q’rDq’
—2(q’r)~q’] 1=2[(D—a)q’Dq’(D+a)]1
=2F’D’FqD’qFD~F’,
(12)
with F=expD~a,
(13)
where a is related to r and q by the following Riccati equation, a~=~a2+q’q~a—2qr.
(14)
Combining eqs. (11) and (12), one obtains A=—FD’FqD’qFD~F’(Dq’Dq’—2rq’).
(15)
In the same way, the operators C and E read C=—2G’D’GrD’GrD~G’,
(16)
E=GD’GrD’rGD’G’(Dr’Dr’—2qr’)
(17)
,
with G=expD~fl,
(18)
while fi is also given by a Riccati equation, fl~=~fl2+r’r~fl—2qr.
(19)
Finally, the left inverse recursion operator for the AKNS hierarchy reads
.(—F 1D ‘FqD ‘qFEr ‘F—’ (Dq— ‘Dq —2G’D’GrD’rGD’G’ —
—
—
2rq ‘)
2F ‘D ‘FqD ‘qFD ‘F G’D’GrD’rGD1G’(Dr’Dr1—2qr’) 1
(20) 272
Volume 179, number 4,5
PHYSICS LETTERS A
16 August 1993
It is easy to prove that 1J given by eq. (20) is also the right inverse operator of ‘~P. At first sight, one can extend the half AKNS hierarchy (1) to a full hierarchy by taking n in (10) being negative and positive integers just as in the formal inverse recursion operator theory. However, there is no other nontrivial equation in the full AKNS hierarchy than that in the half AKNS hierarchy because ‘
s~J~’K
(21)
0=0.
The symmetries of the AKNS hierarchy (1) have been studied widely in the literature [8,9]. The known symmetries are mKo (m=0,l,2,...), (22) Km~I’
t~mI’mT~j~J’m(fltKn....t +XU)fltKn+m....i +‘lm(xu)
(m=0, 1,2,...).
(23)
Now the i~, symmetries can be extended to the negative direction with Km = 0 for m <0. This fact shows us that the nonisospectral AKNS hierarchy [8] can be extended to the full hierarchy u~=r~, (n=0, 1,2,..., m=0, ±1,±2,...),
(24)
though the isospectral AKNS hierarchy is a half one only. The Lie algebra constituted by Km and i~, for the AKNS hierarchy now reads [Km,Ki]0 (m,10, 1,2,...), (m,n=0,l,2,...,1=0,±l,±2,...),
[Km,t7]mKm+i..i
[ta,, il] = (m—1)T~,+,..,
(25) (26)
(n=0, 1,2,..., m, 1=0, ± 1, ±2,...)
,
(27)
with K~=Ofor p
D_1=Jdx,
one can get various new symmetries of models [5—7].However, we fail to get any other new symmetries for the general AKNS hierarchy (1). Finally we will take a special case of (1), say the mKdV hierarchy, r= q and q a real function of x and t. Then the nontrivial equations of (1) read U~=tP2m÷lU
(m,l=0,l,2,...),
(29)
which can be reduced the the scalar equations [8] (30) where Ro=q~,
(31)
~,=_q~+6q2q~,
and k,=_D2+4q~D—’q+4q2.
(32)
At a first glance, m = I in eq. (30) will give a nontrivial integrable model, say the sinh-Gordon equation [3,8]. Actually, this is not true in the real situation. The inverse of ~, [5] is —
‘I’ j~= —q’D(D+2q)’q(D—2q)’D’
(g=exp2D~q) .
= ~gD’g’D’ —
—
(33) 273
Volume 179, number 4,5
PHYSICS LETTERS A
A simple calculation will show that eq. (30) with m =
—
16 August 1993
1 is a trivial equation, (34)
q~=0,
rather than the sinh-Gordon equation. The known sinh-Gordon equation comes from another symmetry hierarchy of the mKdV equation (m = for eq. (30)). In ref. [7], we have given four other hierarchies of new symmetries of the mKdV equation. These symmetries can be rewritten as £~=~j-msinh(q) =~~gD’g1,
~~=~jmcosh(q) ~=~~j~’g’D’g
(m=0,l,2,...), (n=l,2,...) ,
(35) (36)
by combining them linearly. From the first hierarchy of symmetries we get the sinh-Gordon hierarchy q
(m=0, 1,2,...) . (37) The first equation of (37) is just the known sinh-Gordon equation. It is also easy to check that acting ~ on will result in a trivial symmetry instead of u~. In summary, the explicit form of the inverse recursion operator indicates how to get diverse correct new information on the integrable hierarchies. In this Letter, the explicit inverse recursion operator of the AKNS hierarchy is obtained. The nonisospectral AKNS hierarchy is extended to a full hierarchy while the isospectral one is only a half hierarchy. Some types of new infinitely many symmetries can be obtained for special cases, while how to get the new symmetry hierarchies for the general AKNS hierarchy should be studied further. 1=i.~
~~‘)
This work was supported by the Natural Science Foundation of Zhejiang Province and the National Science Foundation of China. The authors would like to thank Professors G. Ni, D. Chen and Dr. H. Ruan for discussions.
References [1]P.J.Olver,J. Math. Phys. 18(1977)1212. [2J B. Fuchssteiner, Nonlinear Analysis TMA 3 (1979) 849. [3] B. Fuchssteiner, Prog. Theor. Phys. 65 (1981) 861. [4]5. Carillo and B. Fuchssteiner, J. Math. Phys. 30 (1989)1606. [5J S. Lou, The fractorization and inverse of the recursion operators for some integrable models, to appear in: Proc. XXI Int. Conf. on Differential geometric methods in theoretical physics, Tianjing, 5—9 June 1992. [6] 5. Lou, Symmetries of KdVequation and four hierarchies ofthe integro-differential KdV equation, preprint NBN-IMP 07/92. [7] 5. Lou, Symmetries, conservation laws and infinite dimensional Lie algebra of the modified KdV equation, Preprint NI3N-IMP 10/ 92. [8]Y. Li andG. Zhu,J. Phys. A 19 (1986) 3713. [9]D.ChenandH.Zheng,J.Phys.A24(1991)377.
274