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The \(N=2\) super \(W_ 4\) algebra and its associated generalized Korteweg- de Vries hierarchies. (English) Zbl 0789.17020
Summary: The \(N=2\) super \(W_ 4\) algebra is constructed as a certain reduction of the second Gel’fand-Dikij bracket on the dual of the Lie superalgebra of \(N = 1\) super pseudodifferential operators. The algebra is put in manifestly \(N = 2\) supersymmetric form in terms of three \(N = 2\) superfields \(\Phi_ i(X)\), with \(\Phi_ i\) being the \(N = 2\) energy momentum tensor and \(\Phi_ 2\) and \(\Phi_ 3\) being conformal spin 2 and 3 superfields, respectively. A search for integrable hierarchies of the generalized Korteweg-de Vries (KdV) variety with this algebra as Hamiltonian structure gives three solutions, exactly the same number as for the \(W_ 2\) (super KdV) and \(W_ 3\) (super Boussinesq) cases.

MSC:
17B68 Virasoro and related algebras
35Q53 KdV equations (Korteweg-de Vries equations)
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.)
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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