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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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##### References:
 [1] Gross, D., Migdal, A.B.: Nonperturbative solution of the Ising model on a random surface. Phys. Rev. Lett.64, 127 (1990); Brézin, E., Kazakov, V.A.: Exactly solvable field theories of closed strings. Phys. Lett.236B, 144–150 (1990); Douglas, M.R., Shenker, S.H.: Strings in less than one dimension. Nucl. Phys.335B, 635–654 (1990); Douglas, M.R.: Strings in less than one dimension and the generalized KdV hierarchies. Phys. Lett.238B, 176–180 (1990) · Zbl 1050.81610 · doi:10.1103/PhysRevLett.64.127 [2] Manin, Yu. I., Radul, A.O.: A supersymmetric extension of the Kadomtsev Petviashvili hierarchy. Commun. Math. Phys.98, 65–77 (1985); Kuperschmidt, B.A.: Super Korteweg-de Vries equations associated to super symetric extension of the Korteweg-de-Vries equation. J. Math. Phys.29, 2499–2506 (1988) · Zbl 0607.35075 · doi:10.1007/BF01211044 [3] Laberge, C.A., Mathieu, P.:N=2 superconformal algebra and integrableO(2) fermionic extensions of the Korteweg-de-Vries equation. Phys. Lett.215B, 718–722 (1988) [4] Oevel, W., Popowicz, Z.: The biHamiltonian structure of fully supersymmetric Korteweg-de-Vries systems. Commun. Math. Phys.139, 441–460 (1991) · Zbl 0742.35063 · doi:10.1007/BF02101874 [5] McArthur, I.N.: On the Integrability of the super-KdV equations, Commun. Math. Phys.148, 177–188 (1992). · Zbl 0753.35086 · doi:10.1007/BF02102371 [6] Gelfand, I.M., Dikii L.A.: Asymptotic behaviour of the resolvent of Sturm Liouville equations and the algebra of the Korteweg-de-Vries equation. Russ. Math. Surv.30:5, 77–113 (1975) · Zbl 0334.58007 · doi:10.1070/RM1975v030n05ABEH001522 [7] Magri, F.: A simple model of the integrable Hamiltonian equation. J. Math. Phys.19, 1156–1162 (1978) · Zbl 0383.35065 · doi:10.1063/1.523777 [8] Gervais, J.-L., Neveu, A.: Dual string spectrum in Polyakov’s quantization (II). Nucl. Phys.209B, 125–145 (1982); Gervais, J.-L.: Infinite family of polynomial functions of the Virasoro generators with vanishing Poisson brackets. Phys. Lett.160B 277–278 (1985) · doi:10.1016/0550-3213(82)90105-5 [9] Olver, P.J.: Applications of Lie Groups to Differential Equations. Berlin Heidelberg New York: Springer 1986 · Zbl 0588.22001
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