On $W_{1+\infty}$ 3-algebra and integrable system

We construct the $W_{1+\infty}$ 3-algebra and investigate the relation between this infinite-dimensional 3-algebra and the integrable systems. Since the $W_{1+\infty}$ 3-algebra with a fixed generator $W^0_0$ in the operator Nambu 3-bracket recovers the $W_{1+\infty}$ algebra, it is natural to derive the KP hierarchy from the Nambu-Poisson evolution equation. For the general case of the $W_{1+\infty}$ 3-algebra, we directly derive the KP and KdV equations from the Nambu-Poisson evolution equation with the different Hamiltonian pairs. We also discuss the connection between the $W_{1+\infty}$ 3-algebra and the dispersionless KdV equations. 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 $W_{1+\infty}$ 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 Schr\"{o}dinger equation and give an application in optical soliton.