Super quasiperiodic wave solutions and asymptotic analysis for $\mathcal N=1$ supersymmetric $\text{KdV}$-type equations

Based on a general multidimensional Riemann theta function and the super Hirota bilinear form, we extend the Hirota method to construct explicit super quasiperiodic (multiperiodic) wave solutions of $\mathcal N=1$supersymmetric KdV-type equations in superspace. We show that the supersymmetric KdV equation does not have an $N$-periodic wave solution with arbitrary parameters for $N\ge2$. In addition, an interesting influencing band occurs among the super quasiperiodic waves under the presence of a Grassmann variable. We also observe that the super quasiperiodic waves are symmetric about this band but collapse along with it. We present a limit procedure for analyzing the asymptotic properties of the super quasiperiodic waves and rigorously show that the super periodic wave solutions tend to super soliton solutions under some “small amplitude” limits.