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

TL;DR

This paper constructs the $W_{1+ abla} ext{infty}$ 3-algebra, explores its connection to integrable systems like KP and KdV hierarchies, and derives related equations using Nambu-Poisson evolution, revealing deep Hamiltonian relationships.

Contribution

It introduces the $W_{1+ abla} ext{infty}$ 3-algebra, derives integrable equations from it, and connects these algebraic structures to physical models such as optical solitons.

Findings

Derived KP and KdV equations from the $W_{1+ abla} ext{infty}$ 3-algebra

Established a link between the algebra and dispersionless KdV equations

Realized the algebra in terms of a complex bosonic field and derived a nonlinear Schrödinger equation

Abstract

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.