On W1+∞W1+∞ 3-algebra and integrable system

We construct the W1+∞W1+∞ 3-algebra and investigate its connection with the integrable systems. Since the W1+∞W1+∞ 3-algebra with a fixed generator W00 in the operator Nambu 3-bracket recovers the W1+∞W1+∞ algebra, it is intrinsically related to the KP hierarchy. For the general case of the W1+∞W1+∞ 3-algebra, we directly derive the KP and KdV equations from the Nambu–Poisson evolution equation with the different Hamiltonian pairs of the KP hierarchy. Due to the Nambu–Poisson evolution equation involves two Hamiltonians, the deep relationship between the Hamiltonian pairs of KP hierarchy is revealed. Furthermore we give a realization of the W1+∞W1+∞ 3-algebra in terms of a complex bosonic field. Based on the Nambu 3-brackets of the complex bosonic field, we derive the (generalized) nonlinear Schrödinger equation and give an application in optical soliton.