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The affine lie algebra C(1)2 and an equation of Hirota and Satsuma
Volume 89A, number 7
PHYSICS LETTERS
31 May 1982
THE AFFINE LIE ALGEBRA C (1) AND AN EQUATION OF HIROTA AND SATSUMA George WILSON Mathematical lnstttute, 2 4 - 2 9 St. Giles, Oxford OX1 3LB, UK
Received 12 February 1982
It is pointed out that an evolutionary system recently introduced by Hirota and Satsuma is nothing but an example of a general theory due to Drinfel'd and Sokolov. Some other, similar, examples are also calculated.
Recently Hirota and Satsuma [ 1] began to study the equation U t = Uxx x + 6 U U x - 1 2 V V x , Vt = - 2 ( V x x x + 3 V V x ) .
(1)
It is an evolutionary system for two unknown functions U(x, t), V(x, t); for V = 0 it reduces to the KdV equation for U. Hirota and Satsuma presented some evidence that this equation might be "integrable", in the sense that the KdV equation is: for example, they found that it had five conservation taws, and conjectured that there would be infinitely many. That is indeed the case: it follows, for example, from the Lax representationL t = [ P , L ] for (1), in whichL is a product of two Schrt3dinger operators with potentials U-+ V (see remark t below). The main purpose of this note, however, is to point out that eq. (1) is an example of a general construction of Drinfel'd and Sokolov [2]. This construction involves the affine (often called Kac-Moody, or Euclidean) Lie algebras. A convenient list of these algebras and their Dynkin diagrams can be found in ref. [3 ], p. 503. (We recall that, like finite dimensional simple Lie algebras, the affine algebras are determined up to isomorphism by their Dynkin diagrams.) Drinfel'd and Sokolov show how to construct (i) for each affine algebra, a hierarchy of modified KdV ,~MKdV) equations (reducing to the usual one for A~I)); (ii) for each finite dimensional semi. simple Lie algebra, a Miura transformation (reducing to the usual one [4] for A1); (iii) for each affine alge332
bra q and each vertex c of its Dynkin diagram F, a hierarchy of KdV equations, related to the MKdV equations for ~ by a Miura transformation of type corresponding to the finite dimensional algebra with Dynkin diagram I' - {c]. (We get the usual KdV hierarchy from either of the two vertices of the diagram for A~I)). We are going to calculate the simplest nontrivial equation of the KdV hierarchy associated with the middle vertex of the Dynkin diagram ~==o==~==< for the algebra C(1). It will turn out to be eq. (1). While we are about it, we shall also calculate the similar equations for the algebras A(2) and D(a2); their Dynkin diagrams are c==)==c==6==o and o~o-~-~--~-o, respectively. To perform these calculations, it is convenient to use the connexion with the two dimensional Toda lattice (2DTL) equations [2,5,6]. The 2DTL equations associated with the algebras o~2(1) ' A "'4(2) an,~ ~ D (2) 3 all take the form Rxt = Ae#R _ BeR-S, Sxt = B e R - s + Ce ~ s .
(2)
Here A, B and C are arbitrary non-zero constants,
and (~, t~) is (2, -2) for C(21),(1,-2) for A(42), and (1, - I ) for D(32). Set r = R x, s = S x , and assign degrees to differential polynomials in (r, s) so that deg r = deg s = 1, and x-differentiation increases degree by one. Then we know [2,5,6] that eq. (2) has infinitely many conserved densities that are homogeneous differential polynomials in (r, s). (For C(1) and D (2) there is one 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
Volume 89A, number 7
PHYSICS LETTERS
o f every even degree; for A(2) there is one of every even
degree not congruent to 1 mod 5.) After a straightforward calculation, the conserved density of degree 4 is found to be H = r 4 + cr2s 2 + s 4
-
(2c/~)rxS2 -- (2c/ot)r2s x
-
+ (4c/al3)rxS x + (4/~2)(rx) 2 + ( 4 / a 2 ) ( s x ) 2 ,
(3)
where the constant c is - 6 for C(1),~- 3 for A(2), and - 2 for D(a2). The simplest non-trivial MKdV equation associated with one of these algebras can be written in the haniltonian form rt = a6H/6r,
st = a S H / 6 s ,
(4)
where a = a/bx, and H is given by 0 ) . We are interested in the KdV hierarchy associated with the middle vertex of the Dynkin diagram. Removing this vertex gives the disconnected diagram o o , which shows that the Miura transformation just splits into two standard ones. Indeed, if we set u
=
(2[l~)r x -- r 2,
o
=
(2/ot)s x -- s 2 ,
(5)
then we have H = u 2 + c u r +0 2
(modulo x-derivatives), so that the desired KdV equations for (u, v) implied by (4) take the form --u t = [(4//32)a 3 + 4ua + 2 U x ] ~ H / ~ u , -1)t = [(4/ot2)a 3 + 4on + 2 O x ] S H / S v .
(6)
I leave it to the reader to write out all these equations explicitly; however, let us note that in the cases C(21) and D(32)the KdV equations (6) take a very simple form when written in terms of the variables (U, V) defined by u=U+
V,
v=U-
V.
(7)
For C(21)we get (after rescaling t) eq. (1). For D(32)we get the system Ut = 3 V V x ,
Vt = 2 V x x x + 2 U V x + U x V .
(8)
The properties of all these equations can now be read off from the general theory (2). For example, all the conserved densities for eq. (2) can be written as differential polynomials in (u, o), and are then also conserved densities for eq. (6).
31 May 1982
Remarks
1. We have been using the hamiltonian forms of eqs. (4) and (6). However, the general theory [2] also provides Lax (or zero curvature) representations for them. Eq. (1), for example, has a Lax representation L t = [ P , L ] in which L = (a 2 + u)(a 2 + v), P = 4a 3 + 6 u a + 3 ( u - 2 V ) x [recall that (u, v) and (U, V) are related by (7)]. For eq. (8), We can take L = (a 3 + u a + ½ux)(a 3 + oa -P= a 3 + ~
+½(u-
+½Vx),
3v) x .
(Different Lax representationsfor eqs. (1)and (8)are given in ref. [2] .) 2. The fact that for V = 0 eq. (1) zedu~cesto the (usual) KdV equation for U is true for the whole hierarchy. Indeed, for V = 0 the operator L in the Lax representation for (1)just becomes (a 2 + U)2, which obviously gives the same hierarchy as a 2 + u. 3. It is not hard to fred all values of (a, ~) such that eq. (2)has a conserved density of degree 4 [necessarily of the form (3)]. Apart from the three cases already considered, there is the possibility (2, - 1 ) , which is equivalent to (1, - 2 ) . Also, ifc = 2, then the expression (3) will be a conserved density for eq. (2) provided that (1/~) - (1/~) = 1. But I know of no reason to expect these eq. (2) to be integrable. 4. I guessed that eq. (1) was connected with an affine algebra after seeing some calculations of Kupershrnidt (7); I am greatly indebted to Kupershmidt for showing me these calculations. It was he who first noticed that eq. (1) is related to eq. (4) by the Miura transformation (5). It should be said, however, that this would by itself not be strong evidence of integrability: indeed, we have that much for all values of the parameters (a,/3, c), but there is no reason to expect the eqs. (4) and (6) to be integrable except for the special values that we have been considering. I hope tha t the examples presented in this note will encourage some readers to study the wonderful paper of Drinfel'd and Sokolov (2). • 333
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PHYSICS LETTERS
References [1] R. Hirota and J. Satsuma, Phys. Lett. 85A (1981) 407. [2] V.G. Drinfel'd and V.V. Sokolov, Dokl. Akad. Nauk SSSR 258 (1981) 11; Soy. Math. Dokl. 23 (1981) 457. [3] S. Helgason, Differential geometry, Lie groups and symmetry spaces (Academic Press, New York, 1978).
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31 May 1982
[4] R.M. Miura, J. Math. Phys. 9 (1968) 1202. [5 ] B.A. Kupershmidt andG. Wilson, Commun. Math. Phys. 81 (1981) 189. [6] G. Wilson, Ergodic theory and dynamical systems, to be published. [7] B.A. Kupershmidt, private communication.
PHYSICS LETTERS
31 May 1982
THE AFFINE LIE ALGEBRA C (1) AND AN EQUATION OF HIROTA AND SATSUMA George WILSON Mathematical lnstttute, 2 4 - 2 9 St. Giles, Oxford OX1 3LB, UK
Received 12 February 1982
It is pointed out that an evolutionary system recently introduced by Hirota and Satsuma is nothing but an example of a general theory due to Drinfel'd and Sokolov. Some other, similar, examples are also calculated.
Recently Hirota and Satsuma [ 1] began to study the equation U t = Uxx x + 6 U U x - 1 2 V V x , Vt = - 2 ( V x x x + 3 V V x ) .
(1)
It is an evolutionary system for two unknown functions U(x, t), V(x, t); for V = 0 it reduces to the KdV equation for U. Hirota and Satsuma presented some evidence that this equation might be "integrable", in the sense that the KdV equation is: for example, they found that it had five conservation taws, and conjectured that there would be infinitely many. That is indeed the case: it follows, for example, from the Lax representationL t = [ P , L ] for (1), in whichL is a product of two Schrt3dinger operators with potentials U-+ V (see remark t below). The main purpose of this note, however, is to point out that eq. (1) is an example of a general construction of Drinfel'd and Sokolov [2]. This construction involves the affine (often called Kac-Moody, or Euclidean) Lie algebras. A convenient list of these algebras and their Dynkin diagrams can be found in ref. [3 ], p. 503. (We recall that, like finite dimensional simple Lie algebras, the affine algebras are determined up to isomorphism by their Dynkin diagrams.) Drinfel'd and Sokolov show how to construct (i) for each affine algebra, a hierarchy of modified KdV ,~MKdV) equations (reducing to the usual one for A~I)); (ii) for each finite dimensional semi. simple Lie algebra, a Miura transformation (reducing to the usual one [4] for A1); (iii) for each affine alge332
bra q and each vertex c of its Dynkin diagram F, a hierarchy of KdV equations, related to the MKdV equations for ~ by a Miura transformation of type corresponding to the finite dimensional algebra with Dynkin diagram I' - {c]. (We get the usual KdV hierarchy from either of the two vertices of the diagram for A~I)). We are going to calculate the simplest nontrivial equation of the KdV hierarchy associated with the middle vertex of the Dynkin diagram ~==o==~==< for the algebra C(1). It will turn out to be eq. (1). While we are about it, we shall also calculate the similar equations for the algebras A(2) and D(a2); their Dynkin diagrams are c==)==c==6==o and o~o-~-~--~-o, respectively. To perform these calculations, it is convenient to use the connexion with the two dimensional Toda lattice (2DTL) equations [2,5,6]. The 2DTL equations associated with the algebras o~2(1) ' A "'4(2) an,~ ~ D (2) 3 all take the form Rxt = Ae#R _ BeR-S, Sxt = B e R - s + Ce ~ s .
(2)
Here A, B and C are arbitrary non-zero constants,
and (~, t~) is (2, -2) for C(21),(1,-2) for A(42), and (1, - I ) for D(32). Set r = R x, s = S x , and assign degrees to differential polynomials in (r, s) so that deg r = deg s = 1, and x-differentiation increases degree by one. Then we know [2,5,6] that eq. (2) has infinitely many conserved densities that are homogeneous differential polynomials in (r, s). (For C(1) and D (2) there is one 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
Volume 89A, number 7
PHYSICS LETTERS
o f every even degree; for A(2) there is one of every even
degree not congruent to 1 mod 5.) After a straightforward calculation, the conserved density of degree 4 is found to be H = r 4 + cr2s 2 + s 4
-
(2c/~)rxS2 -- (2c/ot)r2s x
-
+ (4c/al3)rxS x + (4/~2)(rx) 2 + ( 4 / a 2 ) ( s x ) 2 ,
(3)
where the constant c is - 6 for C(1),~- 3 for A(2), and - 2 for D(a2). The simplest non-trivial MKdV equation associated with one of these algebras can be written in the haniltonian form rt = a6H/6r,
st = a S H / 6 s ,
(4)
where a = a/bx, and H is given by 0 ) . We are interested in the KdV hierarchy associated with the middle vertex of the Dynkin diagram. Removing this vertex gives the disconnected diagram o o , which shows that the Miura transformation just splits into two standard ones. Indeed, if we set u
=
(2[l~)r x -- r 2,
o
=
(2/ot)s x -- s 2 ,
(5)
then we have H = u 2 + c u r +0 2
(modulo x-derivatives), so that the desired KdV equations for (u, v) implied by (4) take the form --u t = [(4//32)a 3 + 4ua + 2 U x ] ~ H / ~ u , -1)t = [(4/ot2)a 3 + 4on + 2 O x ] S H / S v .
(6)
I leave it to the reader to write out all these equations explicitly; however, let us note that in the cases C(21) and D(32)the KdV equations (6) take a very simple form when written in terms of the variables (U, V) defined by u=U+
V,
v=U-
V.
(7)
For C(21)we get (after rescaling t) eq. (1). For D(32)we get the system Ut = 3 V V x ,
Vt = 2 V x x x + 2 U V x + U x V .
(8)
The properties of all these equations can now be read off from the general theory (2). For example, all the conserved densities for eq. (2) can be written as differential polynomials in (u, o), and are then also conserved densities for eq. (6).
31 May 1982
Remarks
1. We have been using the hamiltonian forms of eqs. (4) and (6). However, the general theory [2] also provides Lax (or zero curvature) representations for them. Eq. (1), for example, has a Lax representation L t = [ P , L ] in which L = (a 2 + u)(a 2 + v), P = 4a 3 + 6 u a + 3 ( u - 2 V ) x [recall that (u, v) and (U, V) are related by (7)]. For eq. (8), We can take L = (a 3 + u a + ½ux)(a 3 + oa -P= a 3 + ~
+½(u-
+½Vx),
3v) x .
(Different Lax representationsfor eqs. (1)and (8)are given in ref. [2] .) 2. The fact that for V = 0 eq. (1) zedu~cesto the (usual) KdV equation for U is true for the whole hierarchy. Indeed, for V = 0 the operator L in the Lax representation for (1)just becomes (a 2 + U)2, which obviously gives the same hierarchy as a 2 + u. 3. It is not hard to fred all values of (a, ~) such that eq. (2)has a conserved density of degree 4 [necessarily of the form (3)]. Apart from the three cases already considered, there is the possibility (2, - 1 ) , which is equivalent to (1, - 2 ) . Also, ifc = 2, then the expression (3) will be a conserved density for eq. (2) provided that (1/~) - (1/~) = 1. But I know of no reason to expect these eq. (2) to be integrable. 4. I guessed that eq. (1) was connected with an affine algebra after seeing some calculations of Kupershrnidt (7); I am greatly indebted to Kupershmidt for showing me these calculations. It was he who first noticed that eq. (1) is related to eq. (4) by the Miura transformation (5). It should be said, however, that this would by itself not be strong evidence of integrability: indeed, we have that much for all values of the parameters (a,/3, c), but there is no reason to expect the eqs. (4) and (6) to be integrable except for the special values that we have been considering. I hope tha t the examples presented in this note will encourage some readers to study the wonderful paper of Drinfel'd and Sokolov (2). • 333
Volume 89A, number 7
PHYSICS LETTERS
References [1] R. Hirota and J. Satsuma, Phys. Lett. 85A (1981) 407. [2] V.G. Drinfel'd and V.V. Sokolov, Dokl. Akad. Nauk SSSR 258 (1981) 11; Soy. Math. Dokl. 23 (1981) 457. [3] S. Helgason, Differential geometry, Lie groups and symmetry spaces (Academic Press, New York, 1978).
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31 May 1982
[4] R.M. Miura, J. Math. Phys. 9 (1968) 1202. [5 ] B.A. Kupershmidt andG. Wilson, Commun. Math. Phys. 81 (1981) 189. [6] G. Wilson, Ergodic theory and dynamical systems, to be published. [7] B.A. Kupershmidt, private communication.