Super Riemann theta function periodic wave solutions and rational characteristics for a supersymmetric KdV–Burgers equation

Using a multidimensional super Riemann theta function, we propose two key theorems for explicitly constructing multiperiodic super Riemann theta function periodic wave solutions of supersymmetric equations in the superspace $\mathbb{R}_{\Lambda}^{N+1,M}$, which is a lucid and direct generalization of the super-Hirota–Riemann method. Once a supersymmetric equation is written in a bilinear form, its super Riemann theta function periodic wave solutions can be directly obtained by using our two theorems. As an application, we present a supersymmetric Korteweg–de Vries–Burgers equation. We study the limit procedure in detail and rigorously establish the asymptotic behavior of the multiperiodic waves and the relations between periodic wave solutions and soliton solutions. Moreover, we find that in contrast to the purely bosonic case, an interesting phenomenon occurs among the super Riemann theta function periodic waves in the presence of the Grassmann variable. The super Riemann theta function periodic waves are symmetric about the band but collapse along with the band. Furthermore, the results can be extended to the case $N>2$; here, we only consider an existence condition for an $N$-periodic wave solution of a general supersymmetric equation.