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We devise a lucid and straightforward way for explicitly constructingquasi-periodic wave solutions (also called multi-periodic wave solutions) ofsupersymmetric equations in superspace $\mathbb{R}_\Lambda^{2,1}$ overtwo-dimensional Grassmann algebra $G_1(\sigma)$. Once a nonlinear equation iswritten in a bilinear form, its quasi-periodic wave solutions can be directlyobtained by using a formula. Moreover, properties of these solutions areinvestigated in detail by analyzing their structures, plots and asymptoticbehaviors. The relations between the quasi-periodic wave solutions and solitonsolutions are rigorously established. It is shown that the soliton solutionscan be obtained only as limiting cases of the quasi-periodic wave solutionsunder small amplitude limits in superspace $\mathbb{R}_\Lambda^{2,1}$. We findthat, in contrast to the purely bosonic case, there is an interestinginfluencing band occurred among the quasi-periodic waves under the presence ofthe Grassmann variable. The quasi-periodic waves are symmetric about the bandbut collapse along with the band. Furthermore, the amplitudes of thequasi-periodic waves increase as the waves move away from the band. Theefficiency of our proposed method can be demonstrated on a class variety ofsupersymmetric equations such as those considered in this paper,$\mathcal{N}=1$ supersymmetric KdV, Sawada-Kotera-Ramani and Ito's equations,as well as $\mathcal{N}=2$ supersymmetric KdV equation.

Bilinear approach to the quasi-periodic wave solutions of supersymmetric equations in superspace