N=4 SUPER KDV HIERARCHY IN N=4 AND N=2 SUPERSPACES

We present the results of further analysis of the integrability proper ties of the N=4 supersymmetric Korteweg-de Vries (KdV) equation deduce d earlier by two of us [F. Delduc and E. Ivanov, Phys. Lett. B 309, 31 2 (1993)] as a Hamiltonian flow on N=4 SU(2) superconformal algebra in the harmonic N=4 superspace. To make this equation and the relevant H amiltonian structures more tractable, we reformulate it in the ordinar y N=4 and further in N=2 superspaces. In N=2 superspace it is represen ted by a coupled system of evolution equations for a general N=2 super field and two chiral and antichiral superfields, and involves two inde pendent real parameters, a and b. We construct a few first bosonic con served charges in involution, of dimensions from 1 to 6, and show that they exist only for the following choices of the parameters: (i) a=4, b=0; (ii) a=-2, b=-6; (iii) a=-2, b=6. The same values are needed for the relevant evolution equations, including N=4 KdV itself, to be bi- Hamiltonian. We demonstrate that the above three options are related v ia SU(2) transformations and actually amount to the SU(2) covariant in tegrability condition found in the harmonic superspace approach. Our r esults provide a strong evidence that the unique N=4 SU(2) super KdV h ierarchy exists. Upon reduction to N=2 KdV, the above three possibilit ies cease to be equivalent. 0 They give rise to the a=4 and a=-2 N=2 K dV hierarchies, which thus prove to be different truncations of the si ngle N=4 SU(2) KdV one. (C) 1996 American Institute of Physics.