W1+∞ 3-algebra and the higher-order nonlinear Schrödinger equations in optical fiber

In terms of W1+∞ 3-algebra, the generalized Nambu-Poisson evolution equations involving two Hamiltonians are constructed. Given the different Hamiltonian pairs, the higher-order nonlinear Schrödinger equations are obtained. Meanwhile, on the basis of the Maxwell equations, the higher-order nonlinear Schrödinger equations in optical fiber are derived in detail. Furthermore, the relations between the higher-order nonlinear Schrödinger equations in optical fiber and the higher-order nonlinear Schrödinger equations based on the W1+∞ 3-algebra are investigated. The bright soliton, dark soliton and the periodic traveling wave solutions of the higher-order nonlinear Schrödinger equation based on the W1+∞ 3-algebra are also derived.