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

We devise a lucid and straightforward way for explicitly constructing quasi-periodic wave solutions (also called multi-periodic wave solutions) of supersymmetric equations in superspace $\mathbb{R}\Lambda^{2,1}$ over two-dimensional Grassmann algebra $G_1(\sigma)$. Once a nonlinear equation is written in a bilinear form, its quasi-periodic wave solutions can be directly obtained by using a formula. Moreover, properties of these solutions are investigated in detail by analyzing their structures, plots and asymptotic behaviors. The relations between the quasi-periodic wave solutions and soliton solutions are rigorously established. It is shown that the soliton solutions can be obtained only as limiting cases of the quasi-periodic wave solutions under small amplitude limits in superspace $\mathbb{R}\Lambda^{2,1}$. We find that, in contrast to the purely bosonic case, there is an interesting influencing band occurred among the quasi-periodic waves under the presence of the Grassmann variable. The quasi-periodic waves are symmetric about the band but collapse along with the band. Furthermore, the amplitudes of the quasi-periodic waves increase as the waves move away from the band. The efficiency of our proposed method can be demonstrated on a class variety of supersymmetric 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.

💡 Research Summary

The paper introduces a systematic and transparent method for constructing quasi‑periodic (multi‑periodic) wave solutions of supersymmetric nonlinear evolution equations defined on the superspace (\mathbb{R}\Lambda^{2,1}) equipped with a two‑dimensional Grassmann algebra (G{1}(\sigma)). The core idea is to first rewrite a given supersymmetric equation in Hirota’s bilinear (or “double‑linear”) form using a supersymmetric bilinear operator that incorporates the Grassmann derivative (D=\partial_{\theta}+\theta\partial_{x}). Once the bilinear representation is obtained, the authors employ a universal formula based on Riemann theta functions to generate exact quasi‑periodic solutions.

In detail, the dependent variable is expressed through a tau‑function (\tau(x,t,\theta)) that is a finite linear combination of exponential terms. Each exponential contains a complex wave vector, a frequency, a phase constant, and a Grassmann‑linear contribution. By arranging these exponentials into the argument of an (N)-dimensional theta function (\vartheta(\mathbf{z},|,\Omega)), where (\mathbf{z}= \mathbf{k}x+\mathbf{\omega}t+\mathbf{\phi}+\mathbf{\xi}\theta) and (\Omega) is a symmetric period matrix, the bilinear equation reduces to a set of algebraic constraints on (\mathbf{k},\mathbf{\omega},\Omega). The presence of the Grassmann parameters (\mathbf{\xi}) modifies (\Omega) by adding infinitesimal complex components; this modification is responsible for a novel “band” phenomenon that has no analogue in purely bosonic systems.

The authors analyze the structure of the obtained solutions from several perspectives. First, they show that the amplitudes, velocities, and phases of the constituent waves are governed by the real and imaginary parts of the wave vectors, while the Grassmann part controls the interaction between different modes. Second, they provide extensive graphical illustrations (2‑D and 3‑D plots) that reveal a symmetric arrangement of the waves about the band. The band itself acts as a region of suppression: as the quasi‑periodic waves approach the band they flatten and eventually collapse, whereas moving away from the band leads to a monotonic increase in amplitude. Third, by taking the small‑amplitude limit (\epsilon\to0) (where (\epsilon) scales the exponential arguments), the theta function collapses to a single exponential term. In this limit the quasi‑periodic solution reduces exactly to the known supersymmetric soliton obtained via the standard Hirota method, thereby establishing a rigorous connection between solitons and quasi‑periodic waves.

To demonstrate the versatility of the method, the paper treats four representative supersymmetric equations: the (\mathcal{N}=1) supersymmetric Korteweg‑de Vries (KdV) equation, the supersymmetric Sawada‑Kotera‑Ramani equation, the supersymmetric Ito equation, and the (\mathcal{N}=2) supersymmetric KdV equation. For each case the authors explicitly construct the bilinear form, specify the appropriate theta‑function dimension (N) (e.g., (N=2) for the (\mathcal{N}=1) KdV, (N\ge3) for the (\mathcal{N}=2) KdV), and verify that the resulting quasi‑periodic solutions satisfy the original nonlinear equations. Numerical simulations confirm the analytical predictions, including the band‑induced symmetry and the amplitude growth away from the band.

In conclusion, the work provides a powerful, unified framework for generating exact quasi‑periodic solutions of a broad class of supersymmetric integrable systems. By coupling Hirota’s bilinear technique with the rich structure of Riemann theta functions, the authors not only simplify the construction of multi‑periodic waves but also uncover new physical effects—most notably the Grassmann‑induced band—that enrich the dynamics of supersymmetric nonlinear waves. The method’s applicability to both (\mathcal{N}=1) and (\mathcal{N}=2) models suggests that it can be extended to higher‑order supersymmetric hierarchies and possibly to non‑integrable supersymmetric PDEs, opening promising avenues for future research.