Quasiperiodic wave solutions of supersymmetric KdV equation in superspace

A direct and unifying scheme for explicitly constructing quasiperiodic wave solutions (multiperiodic wave solutions) of supersymmetric KdV equation in a superspace is proposed. The scheme is based on the concept of super Hirota forms and on the use of super Riemann theta functions. In contrast to ordinary KdV equation with purely bosonic field, some new phenomena on super quasiperiodic waves occur in the supersymmetric KdV equation with the fermionic field. For instance, it is shown that the supersymmetric KdV equation does not possess an N-periodic wave solution for N≥ 2 for arbitrary parameters. It is further observed that there is an influencing band occurred among the quasiperiodic waves under the presence of the Grassmann variable. The quasiperiodic waves are symmetric about the band but collapse along with the band. In addition, the relations between the quasiperiodic wave solutions and soliton solutions are rigorously established. It is shown that quasiperiodic wave solution convergence to the soliton solutions under certain conditions and small amplitude limit. © 2010 by the Massachusetts Institute of Technology.