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A super Degasperis-Procesi equation and related integrable systems
Binfang Gao1, Kai Tian2, Qing Ping Liu2
1Faculty of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006, People's Republic of China.
Proceedings. Mathematical, Physical, and Engineering Sciences
|March 1, 2021
Summary
A new super Degasperis-Procesi (DP) equation is proposed and linked to super Kaup-Kupershmidt (KK) and super Boussinesq hierarchies. This study establishes bi-Hamiltonian structures and infinite conservation laws for these integrable systems.
Area of Science:
- Integrable systems
- Mathematical physics
- Nonlinear partial differential equations
Background:
- The study of integrable systems is crucial for understanding complex nonlinear phenomena.
- Super extensions of classical integrable equations offer new mathematical structures and applications.
- The Degasperis-Procesi (DP) and Kaup-Kupershmidt (KK) equations are significant nonlinear evolution equations.
Purpose of the Study:
- To propose a super Degasperis-Procesi (DP) equation based on a 4x4 matrix spectral problem.
- To establish connections between the super DP equation and super Kaup-Kupershmidt (KK) and super Boussinesq hierarchies.
- To investigate the bi-Hamiltonian structures and conservation laws of these super integrable systems.
Main Methods:
- Formulation of a super DP equation using a 4x4 matrix spectral problem.
- Application of a reciprocal transformation to relate different super hierarchies.
- Establishment of bi-Hamiltonian structures for super Boussinesq and super KK hierarchies.
- Derivation of conservation laws for the super DP equation and its flows.
Main Results:
- A novel super DP equation is proposed.
- The super DP equation is shown to be related to the first negative flow of a super KK hierarchy, a reduction of a super Boussinesq hierarchy.
- The bi-Hamiltonian structure of the super Boussinesq hierarchy is established, leading to a Hamiltonian structure for the super KK hierarchy.
- A bi-Hamiltonian representation for the super DP equation is constructed.
- Infinitely many conservation laws are derived for the super DP equation and its positive flow.
Conclusions:
- The proposed super DP equation is integrable and connected to other important super integrable hierarchies.
- The study provides a comprehensive analysis of the Hamiltonian structures and conservation laws.
- This work contributes to the broader understanding of super integrable nonlinear evolution equations.
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