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$$W_{1+\infty}3$$-algebra and the higher-order nonlinear Schrödinger equations in optical fiber. (English) Zbl 07263838
Summary: In terms of $$W_{1+\infty}3$$-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 $$W_{1+\infty}3$$-algebra are investigated. The bright soliton, dark soliton and the periodic traveling wave solutions of the higher-order nonlinear Schrödinger equation based on the $$W_{1+\infty}3$$-algebra are also derived.
##### MSC:
 00 General and overarching topics; collections
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