In terms of the operator Nambu 3-bracket and the Lax pair (L, B n ) of the KP hierarchy, we propose the generalized Lax equation with respect to the Lax triple (L, B n , B m ). The intriguing results are that we derive the KP equation and another integrable equation in the KP hierarchy from the generalized Lax equation with the different Lax triples (L, B n , B m ). Furthermore we derive some no integrable evolution equations and present their single soliton solutions.
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.
We investigate the super high-order Virasoro 3-algebra. By applying the appropriate scaling limits on the generators, we obtain the super w ∞ 3-algebra which satisfies the generalized fundamental identity condition. We also define a super Nambu-Poisson bracket which satisfies the generalized skewsymmetry, Leibniz rule and fundamental identity. By means of this super Nambu-Poisson bracket, the realization of the super w ∞ 3-algebra is presented.
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