超空间中超对称方程拟周期波解的双线性逼近
Bilinear approach to the quasi-periodic wave solutions of supersymmetric equations in superspace
摘要：
本文给出了一种简洁明了的方法来构造二维Grassmann代数$G_1(σ)$上超空间$\mathbb{R}_Λ^{2,1}$中超对称方程的拟周期波解（又称多周期波解）.一旦一个非线性方程被写成双线性形式，它的拟周期波解就可以通过公式直接得到。通过分析解的结构、图和渐近性态，详细研究了这些解的性质。严格地建立了准周期波解与孤子解之间的关系。结果表明，孤子解只能作为超空间$\mathbb{R}_Λ^{2,1}$中小振幅极限下准周期波解的极限情形得到。我们发现，与纯玻色子的情况下，有一个有趣的影响带发生在准周期波的格拉斯曼变量的存在下。准周期波关于带对称，但随带坍缩。此外，准周期波的振幅随着波远离频带而增加。我们所提出的方法的有效性可以证明对一类超对称方程，如本文所考虑的，$\mathcal{N}=1$超对称KdV，Sawada-Kotera-Ramani和伊藤的方程，以及$\mathcal{N}=2$超对称KdV方程。
英文摘要：
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 $\mathbb{R}_Λ^{2,1}$ over two-dimensional Grassmann algebra $G_1(σ)$. 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 $\mathbb{R}_Λ^{2,1}$. 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.
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