3 Nonlinear superposition formula for N = 1 supersymmetric KdV Equation

In this paper, we derive a Bäcklund transformation for the supersymmetric Korteweg-de Vries equation. We also construct a nonlinear superposition formula, which allows us to rebuild systematically for the supersymmetric KdV equation the soliton solutions of Carstea, Ramani and Grammaticos. The celebrated Korteweg-de Vries (KdV) equation was extended into super framework by Kupershmidt [3] in 1984. Shortly afterwards, Manin and Radul [7] proposed another super KdV system which is a particular reduction of their general supersymmet-ric Kadomstev-Petviashvili hierarchy. In [8], Mathieu pointed out that the super version of Manin and Radul for the KdV equation is indeed invariant under a space supersymmetric transformation, while Kupershmidt's version does not. Thus, the Manin-Radul's super KdV is referred to the supersymmetric KdV equation. We notice that the supersymmetric KdV equation has been studied extensively in literature and a number of interesting properties has been established. We mention here the infinite conservation laws [8], bi-Hamiltonian structures [10], bilinear form [9][2], Darboux transformation [6]. By the constructed Darboux transformation, Mañas and one of us calculated the soli-ton solutions for the supersymmetric KdV system. This sort of solutions was also obtained by Carstea in the framework of bilinear formalism [1]. However, these solutions are characterized by the presentation of some constraint on soliton parameters. Recently, using super-bilinear operators, Carstea, Ramani and Grammaticos [2] constructed explicitly new two-and three-solitons for the supersymmetric KdV equation. These soliton solutions are interesting since they are free of any constraint on soliton parameters. Furthermore, the fermionic part of these solutions is dressed through the interactions. In addition to the bilinear form approach, Bäcklund transformation (BT) is also a powerful method to construct solutions. Therefore, it is interesting to see if the soliton solutions of Carstea-Ramani-Grammaticos can be constructed by BT approach. In this paper, we first construct a BT for the supersymmetric KdV equation. Then, we derive a nonlinear superposition formula. In this way, the soliton solutions can be produced 1