The Equal-time And Equal-space Poisson Brackets Of The Manakov Equation

The existence of Hamiltonian structures or Poisson brackets and the spectral ma-trix satisfying the r-matrix formula are the basic features of integrable systems.It has proved that almost all soliton equations possess bi-Hamiltonian structures which are differential or differential-integral operators with respect to space variable x.These Poisson brackets are called equal-time Poisson brackets,in which time variable is treat-ed as a parameter.It has recently been found that the nonlinear Schrodinger equation also has an equal-space Poisson bracket.The Manakov equation is a 2-component in-tegrable nonlinear Schrodinger equation with important physical applications.In this thesis,we construct both the equal-time and equal-space Poisson brackets of Manakov equation.The thesis consists of three chapters.In the second chapter,we first study the construction and bi-Hamiltonian structure of vector AKNS hierarchy of equations.Then,through the reduction between potential-s,the zero curvature representation and bi-Hamiltonian structure of Manakov equation are given,and two equal-time Poisson brackets are obtained.In the third chapter,starting from the Lagrange formula of Manakov equation,an equal-time and an equal-space Poisson bracket of the Manakov equation are estab-lished.The equal-time Poisson bracket is one obtained in Chapter 2.The Hamilton equation is the usual time-evolution equation.The equal-space Poisson bracket is new,and the corresponding Hamilton equation is the evolution equation of the space vari-able.The Manakov equation can be expressed as Hamiltonian systems in terms of both Poisson brackets.In the fourth chapter,we prove that the spectral matrices U(x,t,?)and V(x,t,?)of Manakov equation respectively satisfy the same form of r-matrix formula.