Gao Binfang, Tian Kai, Liu Qing Ping
Faculty of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, People's Republic of China.
Department of Mathematics, China University of Mining and Technology, Beijing 100083, People's Republic of China.
Proc Math Phys Eng Sci. 2021 Jan;477(2245):20200780. doi: 10.1098/rspa.2020.0780. Epub 2021 Jan 27.
Based on a 4 × 4 matrix spectral problem, a super Degasperis-Procesi (DP) equation is proposed. We show that under a reciprocal transformation, the super DP equation is related to the first negative flow of a super Kaup-Kupershmidt (KK) hierarchy, which turns out to be a particular reduction of a super Boussinesq hierarchy. The bi-Hamiltonian structure of the super Boussinesq hierarchy is established and subsequently produces a Hamiltonian structure, as well as a conjectured symplectic formulation of the super KK hierarchy via suitable reductions. With the help of the reciprocal transformation, the bi-Hamiltonian representation of the super DP equation is constructed from that of the super KK hierarchy. We also calculate a positive flow of the super DP hierarchy and explain its relations with the super KK equation. Infinitely many conservation laws are derived for the super DP equation, as well as its positive flow.
基于一个4×4矩阵谱问题，提出了一个超德加斯佩里斯 - 普罗塞西（DP）方程。我们表明，在一个互易变换下，超DP方程与超考普 - 库珀施密特（KK）层次结构的第一个负流相关，结果证明它是超布辛涅斯克层次结构的一个特殊约化。建立了超布辛涅斯克层次结构的双哈密顿结构，并通过适当的约化随后产生了一个哈密顿结构以及超KK层次结构的一个推测的辛形式。借助互易变换，从超KK层次结构的双哈密顿表示构造出超DP方程的双哈密顿表示。我们还计算了超DP层次结构的一个正流，并解释了它与超KK方程的关系。为超DP方程及其正流导出了无穷多个守恒律。
