IL NUOVO CIMENTO VOL. 110 A, N. 3 Marzo 1997 NOTE BREVI
An integrable system of coupled non-linear evolution equations
and its supersymmetric version
P. K. ROY(*)
Department of Physics, Presidency College - Calcutta 700 073, India (ricevuto il 2 Gennaio 1997; approvato il 28 Aprile 1997)
Summary. — A set of coupled non-linear wave equations is proposed. We show that
the dynamical system possesses an infinite number of symmetries and conservation laws. The pair can also be written in bi-Hamiltonian form giving rise to an infinite hierarchy of coupled equations each of which is a Hamiltonian system. We also supersymmetrize the equations and write down the fermionic counterparts.
PACS 11.30.Pb – Supersymmetry.
Coupled non-linear waves have been studied by many authors [1-4]. Just as in the case of KdV and similar type equations, coupled systems can also be endowed with symmetries and conservation laws. These in turn may yield a chain of accompanying Hamiltonians. In particular, systems which are embedded in a bi-Hamiltonian structure draw more attention of the physicists because they ensure the existence of an infinite number of independent, commuting conserved quantities.
Historically Hirota and Satsuma [5] (HS) first proposed a coupled KdV equation and examined its symmetry and constants of motion. A year later Ito [6] developed
another system having bi-Hamiltonian structure. These works encouraged
Fuchsteiner [7] to study systematically various possibilities of other couples and their merits and demerits. He concluded that the bi-Hamiltonian formulation of the HS equation does not result out of a Hamiltonian pair. Wilson [8], on the other hand, established the fact that one may obtain different coupled equations from affine algebra associated with the calculations of two-dimensional Toda lattice (2DTL).
Precisely, he found that the HS equation results out of C2( 1 )affine algebra of 2DTL (of
course after rescaling t). More recently, a dispersive generalization of the long water wave equation proposed by Kupershmidt [9] has received much attention. In fact, this system has a tri-Hamiltonian structure [10].
Examples of coupled systems possessing bi-Hamiltonian structure are, however, very few. Search is still in progress to find new integrable systems particularly after the advent of supersymmetric (SUSY) generalization [11-14] of coupled non-linear
equa-(*) E mail: sdpresiHgiascl01.vsnl.net.in
P.K.ROY
334
tions. In this note we propose a new set of coupled wave equations that permits bi-Hamiltonian formulation. We find its symmetry, Hamiltonian structure, recursion operator that yields infinitely many symmetries and conservation laws. We also provide the supersymmetric construction of the equations and give the SUSY equivalent of the system.
To begin with we observe that the following pair of non-linear wave equations:
ut4 6 vv1, vt4 v31 6(uv)1 (1)
remains invariant under the infinitesimal transformations u(t) Ku(t)1el(u, v),
v(t) Kv(t)1em(u, v) indicating that l and m are essentially infinitesimal symmetry of
(1). This is equivalent to say that, if one defines two matrices
A 4NN N 0 6 ¯v 6 ¯v ¯31 6 ¯u N N N , T 4NN N dt 0 0 dt N N N , (2)
then one has the following matrix equation:
NA 2 TNNN N l m N N N 4NN N 0 0 N N N . (3)
Although (1) appears to be very much similar to the system obtained by Wilson [8]
Ut4 3 VV1, Vt4 2 V31 2 UV11 U1V ,
(4)
which corresponds to affine algebra D3( 2 ) associated with the equations for 2DTL, the
set (1) is, however, quite distinct from (4) from the view point of symmetry and conservation laws.
A close look reveals that (1) may be written in matrix form as N N N u v N N Nt 4NN N 0 2(¯v 1v¯) 2(¯v 1v¯) ¯31 2(u¯ 1 ¯u) N N N N N N u v N N N 4NN N 0 ¯ ¯ 0 N N N N N N v21 6 uv 3 v2 N N N . (5)
On defining two skew-symmetric Hamiltonian matrices
B( 1 )4NN N 0 ¯ ¯ 0 N N N and B( 2 )4NN N 0 2(¯v 1v¯) 2(¯v 1v¯) ¯31 2(u¯ 1 ¯u) N N N (6)
(1) can also be written in bi-Hamiltonian form as well: (u , v)t4 B( 2 )dH(n)2 B( 1 )dH(n 11), (7) where dH 4NN N dHOdu dHOdv N N N
is the vector of variational derivative. On further inspection we can draw the other important conclusion that there exists a recursion operator R which may conveniently be defined [15] as
R 4B( 2 )(B( 1 ))21 4NN N 4 v 12v1¯21 ¯21 4 u 1 2 u1¯21 0 4 v 12v1¯21 N N N . (8)
The existence of the recursion operator ensures that the system (1) possesses an infinite series of symmetries. As a result one gets an infinite chain of Hamiltonians. We
AN INTEGRABLE SYSTEM OF COUPLED NON-LINEAR EVOLUTION EQUATIONS ETC. 335
give below the first few members:
.
`
`
/
`
`
´
H( 0 )4 1 2 u dx , H( 1 )4uv dx , H( 2 ) 4g
3 uv2 2 1 2(v1) 2h
dx , H( 3 ) 4(
10 uv3 2 5 v(v1)2)
dx , H( 4 ) 4(
35 uv4 2 35 v2(v1)2)
dx , (9) and so on.Let us now turn our attention to supersymmetric construction of (1). Using the same notations of superfields and covariant derivative as was used in our previous publication [14] we propose the following two Hamiltonians:
.
`
/
`
´
H( 1 ) 4 1 2 dx du[c Df 1f Dc] 4 1 2 dx[ 2 uv 1j1h 2jh1] , H( 2 ) 4dx duk
3 f(Dc)2 2 1 2D 2cD3cl
4 4dxk
3 uv2 2 1 2 (v1) 2 2 6 jh1v 1 1 2h1h2l
(10)as plausible SUSY generalization of H( 1 ), H( 2 ) respectively of the chain (9). Indeed
corresponding to H( 1 )we have N N N f c N N Nt 4
N
N
N
N N N 0 0 0 0 N N N N N N 4 v¯ 12v1 3 h¯ 12h1 3 h¯ 1h1 2v N N N N N N 4 v¯ 12v1 3 h¯ 12h1 3 h¯ 1h1 2v N N N N N N ¯31 4 u¯ 1 2 u1 3 j¯ 12j1 3 j¯ 1j1 2(¯21 u) N N NN
N
N
N
N
N
N N N d H( 1 ) O¯v d H( 1 ) Odh N N N N N N d H( 1 )Odu d H( 1 ) Odj N N NN
N
N
(11)giving rise to the coupled supersymmetric equations N N N f c N N Nt 4NN N 3 D2(c Dc) D6c 13D2(c Df 1f Dc) N N N , (12)
which on simplification yield . / ´ ut4 6 vv12 3 hh2, vt4 v31 6(uv)12 3 j2h 23jh2, jt4 3(hv)1, ht4 h31 3(hu)11 3(jv)1. (13)
P.K.ROY 336 equations as follows: N N N f c N N Nt 4
N
N
N
N N N 0 0 0 0 N N N N N N ¯ 0 0 21 N N N N N N ¯ 0 0 21 N N N N N N 0 0 0 0 N N NN
N
N
N
N
N
N N N d H( 2 )Odv d H( 2 )Odh N N N N N N d H( 2 ) Odu d H( 2 ) Odj N N NN
N
N
(14)giving rise to another SUSY pair in terms of superfields: N N N f c N N Nt 4NN N 6 Dc D2c D6 c 16D2f Dc N N N . (15)
On expanding the covariant derivatives the componentwise equations take up the following form: . / ´ ut4 6 vv1, j14 6 h1v , vt4 v31 6(uv)12 6(jh1)1, ht4 h31 6(jv)1. (16)
To conclude we have proposed a simple coupled system which is completely inte-grable. It possesses an infinite set of symmetries and constants of motion. We have given first few Hamiltonians of the infinite chain which in turn give rise to an infinite hierarchy of coupled equations each of which is a Hamiltonian system. Lastly we have con-sidered supersymmetric generalization of the model. As in the case of SUSY KdV system here also we find two Hamiltonians leading to two different generalized pairs. Comparison with other integrable models will be published in a forthcoming paper.
* * *
The author would like to thank B. BAGCHI for constant encouragement.
R E F E R E N C E S
[1] FADDEEV L. D. and TAKHTAJAN L. A., Hamiltonian Methods in the Theory of Solitons (Springer, Berlin) 1987.
[2] DAS A., Integrable Models (World Scientific, Singapore) 1989.
[3] KUPERSHMIDT B. A. (Editor), Integrable and Superintegrable Systems (World Scientific, Singapore) 1990.
[4] ABLOWITZM. J. and CLARKSONP. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge, New York) 1991.
[5] HIROTAR. and SATSUMAJ., Phys Lett. A, 85 (1981) 407; see also SATSUMAJ. and HIROTAR., Prog. Phys. Soc. Jpn., 51 (1982) 3390.
[6] ITO M., Phys. Lett. A, 91 (1982) 335.
[7] FUCHSSTEINER B., Prog. Theor. Phys., 68 (1982) 1082. [8] WILSON G., Phys. Lett. A, 89 (1982) 332.
[9] KUPERSHMIDT B. A., Commun. Math. Phys., 99 (1985) 51. [10] KUPERSHMIDT B. A., Mech. Res. Commun., 13 (1986) 47. [11] KUPERSHMIDT B. A., J. Math. Phys., 29 (1988) 1990. [12] BRUNELLI J. C. and DAS A., Phys. Lett. B, 337 (1994) 303. [13] PALIT S. and ROYCHOUDHURY A., J. Phys. A, 29 (1996) 2853. [14] ROY P. K. and BAGCHI B., Nuovo Cimento A, 109 (1984) 597. [15] ANTONOWICZ M. and FORDY A. P., Phys. Lett. A, 122 (1984) 95.
