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Gelfand-Dikii analysis for \(N=2\) supersymmetric KdV equations. (English) Zbl 0767.35079
Summary: We generalize the resolvent approach of Gelfand and Dikii to the KdV equation to study the \(N=2\) supersymmetric KdV equations of Laberge and Mathieu. For the associated Lax operators, we study the coincidence limits of the resolvent kernel and its derivatives, and obtain differential equations which they satisfy. These allows us to obtain recursion relations for the analogues of the Gelfand-Dikii polynomials and to obtain a proof of Hamiltonian integrability of the supersymmetric KdV equations. We are able to write the Lax equations for the corresponding hierarchies in terms of these polynomials.
MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
58J70 Invariance and symmetry properties for PDEs on manifolds
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.)
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