Infinite-dimensional 3-algebra and integrable system

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The relation between the infinite-dimensional 3-algebras and the dispersionless KdV hierarchy is investigated. Based on the infinite-dimensional 3-algebras, we derive two compatible Nambu Hamiltonian structures. Then the dispersionless KdV hierarchy follows from the Nambu-Poisson evolution equation given the suitable Hamiltonians. We find that the dispersionless KdV system is not only a bi-Hamiltonian system, but also a bi-Nambu-Hamiltonian system. Due to the Nambu-Poisson evolution equation involving two Hamiltonians, more intriguing relationships between these Hamiltonians are revealed. As an application, we investigate the system of polytropic gas equations and derive an integrable gas dynamics system in the framework of Nambu mechanics.