A fully supersymmetric AKNS theory. (English) Zbl 0852.35127
Summary: We construct a fully supersymmetric (susy) bi-Hamiltonian theory in four superfields, admitting zero curvature and Lax formulation. This theory is an extension of the classical AKNS, which can be recovered as a reduction. Other supersymmetric theories are obtained as reductions of the susy AKNS, namely a nonlinear Schrödinger, a modified KdV and the Manin-Radul KdV. The susy nonlinear Schrödinger hierarchy is related to the one of Roelofs and Kersten; we determine is bi-Hamiltonian and Lax formulation. Finally, we show that the susy KdVs mentioned before are related through a susy Miura map.
Keywords:
supersymmetric bi-Hamiltonian theory; zero curvature; Lax formulation; AKNS; nonlinear Schrödinger; KdV; Manin-Radul KdVReferences:
| [1] | [CMP] Casati, P., Magri, F., Pedroni, M.: Bihamiltonian manifolds and the {\(\tau\)}-function. Proceedings of the 1991 Joint Summer Research Conference on Mathematical aspects of Classical Field Theory. M. Gotai, J. Marsden, V. Moncrief (eds.) in Contemp. Math.132, 213–234 (1992) · Zbl 0791.58050 |
| [2] | [CN] Chowdhury, R.A., Naskar, M.: On the complete integrability of the supersymmetric nonlinear Schrödinger equation. J. Math. Phys.28, 1809–1812 (1987) · Zbl 0641.35074 · doi:10.1063/1.527440 |
| [3] | [Cor] Cornwell, J.: Group theory in Physics. Vol. III. London: Academic Press (1989) · Zbl 0686.17001 |
| [4] | [CP] Casati, P., Pedroni, M.: Drinfeld-Sokolov reduction on a simple Lie algebra from the biHamiltonian point of view. Lett. Math. Phys.25, 89–101 (1992) · Zbl 0767.58019 · doi:10.1007/BF00398305 |
| [5] | [DS] Drinfeld, V., Sokolov, V.: Lie algebras and equations of Korfteweg-de Vries type. J. Sov. Mat.30, 1995–2036 (1985) |
| [6] | [EW] Estabrook, F.B., Wahlquist, W.D.: Prolongation structures of nonlinear evolution equations II. J. Math. Phys.17, 1293–1297 (1976) · Zbl 0333.35064 · doi:10.1063/1.523056 |
| [7] | [FF] Fuchstteiner, B., Fokas, A.S.: Symlectic structures, their Bäcklund transformations and hereditary symmetries. Physica D4, 47–66 (1981) · Zbl 1194.37114 · doi:10.1016/0167-2789(81)90004-X |
| [8] | [FMR] Figueroa-O’Farrill, J., Mas, J., Ramos, E.: Integrability and biHamiltonian structure of the even order sKdV hierarchies. Rev. Mat. Phys.3, 479–501 (1991) · Zbl 0741.35066 · doi:10.1142/S0129055X91000175 |
| [9] | [IK1] Inami, T., Kanno, H.: Lie superalgebraic approach to super Toda lattice and generalized super KdV equations. Commun. Math. Phys.136, 519–542 (1991) · Zbl 0738.35087 · doi:10.1007/BF02099072 |
| [10] | [IK2] Inami, T., Kanno, H.:N=2 super KdV and super-sine Gordon equations based on Lie superalgebraA(1, 1)(1). Nucl. Phys. B359, 201–217 (1991) · Zbl 1098.37538 · doi:10.1016/0550-3213(91)90297-B |
| [11] | [IK3] Inami, T., Kanno, H.:N=2 super-W-algebras and generalizedN=2 super KdV hierarchies based on Lie superalgebras. J. Phys. A25, 3729–3736 (1992) · Zbl 0754.35139 · doi:10.1088/0305-4470/25/13/021 |
| [12] | [IK4] Inami, T., Kanno, H.: GeneralizedN=2 super KdV hierarchies: Lie superalgebraic methods and scalar super Lax formalism. Proceedings of the RIMS Research Project 1991, ”Infinite Analysis,” in Int. J. Mod. Phys. A7, (Suppl. 1 A) 419–447 (1992) · Zbl 0925.35129 |
| [13] | [Kul] Kulish, P.P.: ICTP, Trieste preprint IC/85/39 (1985) |
| [14] | [Kup] Kupershmidt, B.: A super Korteweg-de Vries equation. Phys. Lett.102, A 213–215 (1984) · doi:10.1016/0375-9601(84)90693-5 |
| [15] | [Lei] Leites, D.: Introduction to the theory of supermanifolds. Russ. Math. Surv.35, 1–64 (1980) · Zbl 0462.58002 · doi:10.1070/RM1980v035n01ABEH001545 |
| [16] | [LM] Laberge, C., Mathieu, P.:N=2 superconformal algebra and integrableO(2) fermionic extension of the KdV equations. Phys. Lett.215 B, 718–722 (1988) |
| [17] | [LiM] Liberman, P., Marle, C.M.: Symplectic geometry and analytical mechanics. Dordrecht: Reidel, (1987) |
| [18] | [Mal] 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 |
| [19] | [Ma2] Magri, F.: A geometrical approach to the nonlinear evolution equations. Proceedings of the 1979 Lecce Meeting on ”Nonlinear evolution equations and dynamical systems” M. Boiti, F. Pempinelli, G. Soliani (eds.) Lect., Notes in Phys.120, New York: Springer, pp. 233–263, (1980) |
| [20] | [MaR] Marsden, J.E., Ratiu, T.: Reduction of Poisson Manifolds. Lett. Math. Phys.11, 161–169 (1986) · Zbl 0602.58016 · doi:10.1007/BF00398428 |
| [21] | [Mat] Mathieu, P.: Supersymmetric extension of the Korteweg-de Vries equation. J. Math. Phys.29, 2499–2506 (1988) · Zbl 0665.35076 · doi:10.1063/1.528090 |
| [22] | [MMR] Magri, F., Morosi, C., Ragnisco, O.: Reduction techniques for infinite-dimensional Hamiltonian systems: Some ideas and applications. Commun. Math. Phys.99, 115–140 (1985) · Zbl 0602.58017 · doi:10.1007/BF01466596 |
| [23] | [MP1] Morosi, C., Pizzocchero, L.: On the biHamiltonian structure of the supersymmetric KdV hierarchies: A Lie superalgebraic approach. Commun. Math. Phys.158, 267–288 (1993) · Zbl 0790.35097 · doi:10.1007/BF02108075 |
| [24] | [MP2] Morosi, C., Pizzocchero, L.: On the biHamiltonian interpretation of the Lax formalism. Rev. Math. Phys.7, 389–430 (1995) · Zbl 0842.35107 · doi:10.1142/S0129055X95000177 |
| [25] | [MP3] Morosi, C., Pizzocchero, L.: Osp(3,2) and gl(3,3) supersymmetric KdV hierarchies. Phys. Lett. A185, 241–252 (1994) · Zbl 0941.37531 · doi:10.1016/0375-9601(94)90611-4 |
| [26] | [MP4] Morosi, C., Pizzocchero, L.: On the equivalence of two supersymmetric KdV theories: A biHamiltonian viewpoint. J. Math. Phys.35, 2397–2407 (1994) · Zbl 0805.58010 · doi:10.1063/1.530510 |
| [27] | [MR] Manin, Y., Radul, A.: A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy. Commun. Math. Phys.98, 65–77 (1985) · Zbl 0607.35075 · doi:10.1007/BF01211044 |
| [28] | [OP] Oevel, W., Popowicz, Z.: The biHamiltonian structure of fully supersymmetric Kortewegde Vries systems. Commun. Math. Phys.139, 441–460 (1991) · Zbl 0742.35063 · doi:10.1007/BF02101874 |
| [29] | [Pop] Popowicz, Z.: The extended supersymmetrization of the Nonlinear Schrödinger equation. Preprint 9/6/1994, Univ. of Wroclaw · Zbl 0961.37515 |
| [30] | [RH] Roelofs, G.H.M., van den Hijligenberg, N.W.: Prolongation structures for sypersymmetric equations. J. Phys. A. Math. Gen.23, 5117–5130 (1990) · Zbl 0734.35116 · doi:10.1088/0305-4470/23/22/007 |
| [31] | [RK] Roelofs, G.H.M., Kersten, P.H.M.: Sypersymmetric extensions of the nonlinear Schrödinger equation: symmetries and coverings. J. Math. Phys.33, 2185–2206 (1992) · Zbl 0761.35103 · doi:10.1063/1.529640 |
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