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\(N=4\) super KdV hierarchy in \(N=4\) and \(N=2\) superspaces. (English) Zbl 0865.58028
J. Math. Phys. 37, No. 3, 1356-1381 (1996); erratum ibid. 38, No. 2, 1224 (1997).
Summary: We present the results of further analysis of the integrability properties of the \(N=4\) supersymmetric Korteweg-de Vries (KdV) equation deduced earlier by two of us [F. Delduc and E. Ivanov, Phys. Lett. B 309, No. 3-4, 312-319 (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 Hamiltonian structures more tractable, we reformulate it in the ordinary \(N=4\) and further in \(N=2\) superspaces. In \(N=2\) superspace it is represented by a coupled system of evolution equations for a general \(N=2\) superfield and two chiral and antichiral superfields, and involves two independent real parameters, \(a\) and \(b\). We construct a few first bosonic conserved 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 via SU(2) transformations and actually amount to the SU(2) covariant integrability condition found in the harmonic superspace approach. Our results provide a strong evidence that the unique \(N=4\) SU(2) super KdV hierarchy exists. Upon reduction to \(N=2\) KdV, the above three possibilities cease to be equivalent. They give rise to the \(a=4\) and \(a=-2\) \(N=2\) KdV hierarchies, which thus prove to be different truncations of the single \(N=4\) SU(2) KdV one.

MSC:
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
58A50 Supermanifolds and graded manifolds
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
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