Two Poisson brackets for the N-component coupled nonlinear Schrdinger(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...Two Poisson brackets for the N-component coupled nonlinear Schrdinger(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.展开更多
It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-d...It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds.Thed we apply them to a truncated spectral model of the quasi-geostrophic flow on a cyclic β-plane.展开更多
For any classical Lie algebra , we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers . The universal form of the Lax pair, equations of motion, Hamiltonian as well a...For any classical Lie algebra , we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers . The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets are provided, and explicit examples for with are also given. For all , it is shown that the dynamics of the - and the -Toda chains are natural reductions of that of the -chain, and for , there is also a family of symmetrically reduced Toda systems, the -Toda systems, which are also integrable. In the quantum case, all -Toda systems with 1$' SRC='http://ej.iop.org/images/0253-6102/41/3/339/ctp_41_3_339_12.gif'/> or 1$' SRC='http://ej.iop.org/images/0253-6102/41/3/339/ctp_41_3_339_13.gif'/> describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all -Toda systems survive after quantization.展开更多
基金Supported by National Natural Science Foundation of China under Grant Nos.11271168 and 11671177by the Priority Academic Program Development of Jiangsu Higher Education Institutionsby Innovation Project of the Graduate Students in Jiangsu Normal University
文摘Two Poisson brackets for the N-component coupled nonlinear Schrdinger(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.
文摘It this paper we obtain existence and bifurcation theorems for homoclinic orbits in three-dimeensional,time dependent and independent,perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds.Thed we apply them to a truncated spectral model of the quasi-geostrophic flow on a cyclic β-plane.
基金The project supported in part by National Natural Science Foundation of China
文摘For any classical Lie algebra , we construct a family of integrable generalizations of Toda mechanics labeled a pair of ordered integers . The universal form of the Lax pair, equations of motion, Hamiltonian as well as Poisson brackets are provided, and explicit examples for with are also given. For all , it is shown that the dynamics of the - and the -Toda chains are natural reductions of that of the -chain, and for , there is also a family of symmetrically reduced Toda systems, the -Toda systems, which are also integrable. In the quantum case, all -Toda systems with 1$' SRC='http://ej.iop.org/images/0253-6102/41/3/339/ctp_41_3_339_12.gif'/> or 1$' SRC='http://ej.iop.org/images/0253-6102/41/3/339/ctp_41_3_339_13.gif'/> describe the dynamics of standard Toda variables coupled to noncommutative variables. Except for the symmetrically reduced cases, the integrability for all -Toda systems survive after quantization.
