Equal-Time and Equal-Space Poisson Brackets of the N-Component Coupled NLS Equation**Supported by National Natural Science Foundation of China under Grant Nos. 11271168 and 11671177, by the Priority Academic Program Development of Jiangsu Higher Education Institutions and by Innovation Project of the Graduate Students in Jiangsu Normal University

Abstract

Two Poisson brackets for the N-component coupled nonlinear Schrödinger (NLS) equation are derived by using the variantional principle. The first one is called the equal-time Poisson bracket which does not depend on time but only on the space variable. Actually it is just the usual one describing the time evolution of system in the traditional theory of integrable Hamiltonian systems. The second one is equal-space and new. It is shown that the spatial part of Lax pair with respect to the equal-time Poisson bracket and temporal part of Lax pair with respect to the equal-space Poisson bracket share the same r-matrix formulation. These properties are similar to that of the NLS equation.