Two Kinds Of Finite Dimensional Systems And Their Applications

In this paper,we introduce two kinds of finite dimensional systems and their applications.On the one hand,another form of Lax pair is obtained by transforming the Lax pair of the soliton equations.Then,the same finite dimensional systems in different coordinates are obtained by using the different polynomial expansions of the spectral function in terms of the spectral parameters.On the other hand,the nonlocal finite dimensional integrable systems corresponding to nonlocal soliton equations are studied.This paper can be summarized as the following four aspects.Firstly,based on three different,polynomial expansions of the spectral function corresponding to the spectral problem of the KdV equation,the same finite dimensional Hamiltonian systems in different coordinates are obtained.For two of them,the Hamiltonian systems under noncanonical Poisson structure are obtained.For another,the Hamiltonian system with standard symplectic structure is built.By using the constraints,the differential integral equations satisfied by the potential u are given for N=2.3.It is proved that the solution set of the differential integral equation is invariant under the action of the KdV equation flow.Further,the integrability of the resulting Hamiltonian systems is proved by constructing the generating function of conserved integral.According to Hamilton-Jacobi theory,the action-angle variables and the Jacobi inversion problem are constructed.Here,the inversion object is a special rational function.so the integral of the inversion problem can be expressed as the logarithmic function.Thus,the N soliton solutions of the KdV equation can be obtained.Secondly,the spectral function polynomial expansion method is applied to other soliton systems.The finite dimensional systems and the conserved integrals corresponding to AKNS equation.mKdV equation.Kaup-Newell equation.Dirac equation and nonlinear Schrodinger equation are given.Thirdly,under Lie-Poisson structure,the nonlocal Hamiltonian system corresponding to the reverse space nonlocal NLS equation is studied.According to its Lax representation,the generating function of conserved integral is constructed and the integrability is proved.In the reverse space nonlocal case,the conserved integrals satisfy:Fm*(-x,t)=(-1)m+1Fm(x,t).Furthermore,the separated coordinates for the Lie-Poisson system on the common level set of the Casimir function are used to give the separated equations for the resulting systems.Based on the Hamilton-Jacobi theory,the action-angle variables and the Jacobi inversion problems are introduced.Finally,the Riemannian θ function solutions of nonlocal NLS equation are obtained by inversion.Fourthly,the nonlocal finite dimensional Hamiltonian system under the LiePoisson structure and the integrability corresponding to the reverse space-time nonlocal mKdV equation and the reverse space-time nonlocal NLS-DNLS equation are given respectively.And the conserved integrals satisfy the symmetry:En(-x,-t)=Fm(x,t).