We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and the...We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.展开更多
We study the l^1-stability of a Haxniltonian-preserving scheme, developed in [Jin and Wen, Comm. Math. Sci., 3 （2005）, 285-315], for the Liouville equation with a discontinuous potential in one space dimension. We p...We study the l^1-stability of a Haxniltonian-preserving scheme, developed in [Jin and Wen, Comm. Math. Sci., 3 （2005）, 285-315], for the Liouville equation with a discontinuous potential in one space dimension. We prove that, for suitable initial data, the scheme is stable in the l^1-norm under a hyperbolic CFL condition which is in consistent with the l^1-convergence results established in [Wen and Jin, SIAM J. Numer. Anal., 46 （2008）, 2688-2714] for the same scheme. The stability constant is shown to be independent of the computational time. We also provide a counter example to show that for other initial data, in particular, the measure-valued initial data, the numerical solution may become l^1-unstable.展开更多
In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertibl...In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation, applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoftian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities.展开更多
We study the Ll-error of a Hamiltonian-preserving scheme, developed in [19], for the Liouville equation with a piecewise constant potential in one space dimension when the initial data is given with perturbation error...We study the Ll-error of a Hamiltonian-preserving scheme, developed in [19], for the Liouville equation with a piecewise constant potential in one space dimension when the initial data is given with perturbation errors. We extend the l1-stability analysis in [46] and apply the Ll-error estimates with exact initial data established in [45] for the same scheme. We prove that the scheme with the Dirichlet incoming boundary conditions and for a class of bounded initial data is Ll-convergent when the initial data is given with a wide class of perturbation errors, and derive the Ll-error bounds with explicit coefficients. The convergence rate of the scheme is shown to be less than the order of the initial perturbation error, matching with the fact that the perturbation solution can be l1-unstable.展开更多
A geometric setting for generally nonconservative mechanical systems on fibred manifolds is proposed. Emphasis is put on an explicit formulation of nonholonomic mechanics when an unconstrained Lagrangian system moves ...A geometric setting for generally nonconservative mechanical systems on fibred manifolds is proposed. Emphasis is put on an explicit formulation of nonholonomic mechanics when an unconstrained Lagrangian system moves in a generally non-potential force field depending on time, positions and velocities, and the constraints are nonholonomic, not necessarily linear in velocities. Equations of motion, and the corresponding Harniltonian equations in intrinsic form are given. Regularity conditions are found and a nonholonomic Legendre transformation is proposed, leading to a canonical form of the nonholonomic Hamiltonian equations for nonconservative systems.展开更多
Staring from a new spectral problem,a hierarchy of the generalized Kaup-Newell soliton equations is derived.By employing the trace identity their Hamiltonian structures are also generated.Then,the generalized Kaup-New...Staring from a new spectral problem,a hierarchy of the generalized Kaup-Newell soliton equations is derived.By employing the trace identity their Hamiltonian structures are also generated.Then,the generalized Kaup-Newell soliton equations are decomposed into two systems of ordinary differential equations.The Abel-Jacobi coordinates are introduced to straighten the flows,from which the algebro-geometric solutions of the generalized KaupNewell soliton equations are obtained in terms of the Riemann theta functions.展开更多
This work is concerned with e1-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with pieeewise constant potentials in one space dimension. We provide an analysis much simpler than these in...This work is concerned with e1-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with pieeewise constant potentials in one space dimension. We provide an analysis much simpler than these in literature and obtain the same half-order convergence rate. We formulate the Liouville equation with discretized velocities into a series of linear convection equations with piecewise constant coefficients, and rewrite the numerical scheme into some immersed interface upwind schemes. The e1-error estimates are then evaluated by comparing the derived equations and schemes.展开更多
The bifurcations of solitary waves and kink waves for variant Boussinesq equations are studied by using the bifurcation theory of planar dynamical systems. The bifurcation sets and the numbers of solitary waves and ki...The bifurcations of solitary waves and kink waves for variant Boussinesq equations are studied by using the bifurcation theory of planar dynamical systems. The bifurcation sets and the numbers of solitary waves and kink waves for the variant Boussinesq equations are presented. Several types explicit formulas of solitary waves solutions and kink waves solutions are obtained. In the end, several formulas of periodic wave solutions are presented.展开更多
This paper proposes an equivalent Hamiltonian equations model for the modular multilevel converter-based high-voltage direct-current(MMC-HVDC)transmission system,and constructs an energy function for multi-machine pow...This paper proposes an equivalent Hamiltonian equations model for the modular multilevel converter-based high-voltage direct-current(MMC-HVDC)transmission system,and constructs an energy function for multi-machine power systems with MMC-HVDC transmission lines.The equivalent Hamiltonian equations model is verified to be able to track the power output dynamics of the full model of an MMC-HVDC transmission system.Both theoretical and numerical studies have been undertaken to validate that the energy function proposed for hybrid AC/DC systems satisfies the conditions of an energy function.The work of this paper bridges the gap between the well-developed direct methods of transient stability analysis and power systems with MMC-HVDC transmission lines.展开更多
It is demonstrated that spectral methods can be used to improve the accuracy of numerical solutions obtained by some lower order methods.More precisely,we can use spectral methods to postprocess numerical solutions of...It is demonstrated that spectral methods can be used to improve the accuracy of numerical solutions obtained by some lower order methods.More precisely,we can use spectral methods to postprocess numerical solutions of initial value differential equations.After a few number of iterations(say 3 to 4),the errors can decrease to a few orders of magnitude less.The iteration uses the Gauss-Seidel type strategy,which gives an explicit way of postprocessing.Numerical examples for ODEs,Hamiltonian system and integral equations are provided.They all indicate that the spectral processing technique can be a very useful way in improving the accuracy of the numerical solutions.In particular,for a Hamiltonian system accuracy is only one of the issues;some other conservative properties are even more important for large time simulations.The spectral postprocessing with the coarse-mesh symplectic initial guess can not only produce high accurate approximations but can also save a significant amount of computational time over the standard symplectic schemes.展开更多
In the last years much progress has been achieved in KAM theory concerning bifurcation of quasi-periodic solutions of Hamiltonian or reversible partial differential equations.We provide an overview of the state of the...In the last years much progress has been achieved in KAM theory concerning bifurcation of quasi-periodic solutions of Hamiltonian or reversible partial differential equations.We provide an overview of the state of the art in this field.展开更多
文摘We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.
基金Innovation Project of the Chinese Academy of Sciences grants K5501312S1,K5502212F1,K7290312G7 and K7502712F7NSFC grant 10601062+1 种基金NSF grant DMS-0608720NSAF grant 10676017
文摘We study the l^1-stability of a Haxniltonian-preserving scheme, developed in [Jin and Wen, Comm. Math. Sci., 3 （2005）, 285-315], for the Liouville equation with a discontinuous potential in one space dimension. We prove that, for suitable initial data, the scheme is stable in the l^1-norm under a hyperbolic CFL condition which is in consistent with the l^1-convergence results established in [Wen and Jin, SIAM J. Numer. Anal., 46 （2008）, 2688-2714] for the same scheme. The stability constant is shown to be independent of the computational time. We also provide a counter example to show that for other initial data, in particular, the measure-valued initial data, the numerical solution may become l^1-unstable.
基金supported by the National Natural Science Foundation of China(Grant No.11272050)the Excellent Young Teachers Program of North China University of Technology(Grant No.XN132)the Construction Plan for Innovative Research Team of North China University of Technology(Grant No.XN129)
文摘In this paper, we focus on the construction of structure preserving algorithms for Birkhoffian systems, based on existing symplectic schemes for the Hamiltonian equations. The key of the method is to seek an invertible transformation which drives the Birkhoffian equations reduce to the Hamiltonian equations. When there exists such a transformation, applying the corresponding inverse map to symplectic discretization of the Hamiltonian equations, then resulting difference schemes are verified to be Birkhoftian symplectic for the original Birkhoffian equations. To illustrate the operation process of the method, we construct several desirable algorithms for the linear damped oscillator and the single pendulum with linear dissipation respectively. All of them exhibit excellent numerical behavior, especially in preserving conserved quantities.
文摘We study the Ll-error of a Hamiltonian-preserving scheme, developed in [19], for the Liouville equation with a piecewise constant potential in one space dimension when the initial data is given with perturbation errors. We extend the l1-stability analysis in [46] and apply the Ll-error estimates with exact initial data established in [45] for the same scheme. We prove that the scheme with the Dirichlet incoming boundary conditions and for a class of bounded initial data is Ll-convergent when the initial data is given with a wide class of perturbation errors, and derive the Ll-error bounds with explicit coefficients. The convergence rate of the scheme is shown to be less than the order of the initial perturbation error, matching with the fact that the perturbation solution can be l1-unstable.
基金supported by the Czech Science Foundation (Grant No.GA CˇR 201/09/0981)the Czech-Hungarian Cooperation Programme "Kontakt" (Grant No. MEB041005)the IRSES project ’GEOMECH’ (Grant No. 246981) within the 7th European Community Framework Programme
文摘A geometric setting for generally nonconservative mechanical systems on fibred manifolds is proposed. Emphasis is put on an explicit formulation of nonholonomic mechanics when an unconstrained Lagrangian system moves in a generally non-potential force field depending on time, positions and velocities, and the constraints are nonholonomic, not necessarily linear in velocities. Equations of motion, and the corresponding Harniltonian equations in intrinsic form are given. Regularity conditions are found and a nonholonomic Legendre transformation is proposed, leading to a canonical form of the nonholonomic Hamiltonian equations for nonconservative systems.
基金Supported by the Natural Science Foundation of China(Grant Nos.11547175,11271008)Supported by the Science and Technology Department of Henan Province(No.182102310978)Supported by the Aid Project for the Mainstay Young Teachers in Henan Provincial Institutions of Higher Education of China(Grant Nos.2017GGJS145,2014GGJS-195)
文摘Staring from a new spectral problem,a hierarchy of the generalized Kaup-Newell soliton equations is derived.By employing the trace identity their Hamiltonian structures are also generated.Then,the generalized Kaup-Newell soliton equations are decomposed into two systems of ordinary differential equations.The Abel-Jacobi coordinates are introduced to straighten the flows,from which the algebro-geometric solutions of the generalized KaupNewell soliton equations are obtained in terms of the Riemann theta functions.
文摘This work is concerned with e1-error estimates on a Hamiltonian-preserving scheme for the Liouville equation with pieeewise constant potentials in one space dimension. We provide an analysis much simpler than these in literature and obtain the same half-order convergence rate. We formulate the Liouville equation with discretized velocities into a series of linear convection equations with piecewise constant coefficients, and rewrite the numerical scheme into some immersed interface upwind schemes. The e1-error estimates are then evaluated by comparing the derived equations and schemes.
基金Project supported by the Applied Basic Research Foundations of Sichuan Province of China (No.05JY029-068-2)
文摘The bifurcations of solitary waves and kink waves for variant Boussinesq equations are studied by using the bifurcation theory of planar dynamical systems. The bifurcation sets and the numbers of solitary waves and kink waves for the variant Boussinesq equations are presented. Several types explicit formulas of solitary waves solutions and kink waves solutions are obtained. In the end, several formulas of periodic wave solutions are presented.
基金supported in part by the National Natural Science Foundation of China under Grant No.51807067the State Key Program of National Natural Science Foundation of China under Grant No.U1866210,Young Elite Scientists Sponsorship Program by CSEE under Grant No.CSEE-YESS-2018the Fundamental Research Funds for the Central Universities of China under Grant No.2018MS77。
文摘This paper proposes an equivalent Hamiltonian equations model for the modular multilevel converter-based high-voltage direct-current(MMC-HVDC)transmission system,and constructs an energy function for multi-machine power systems with MMC-HVDC transmission lines.The equivalent Hamiltonian equations model is verified to be able to track the power output dynamics of the full model of an MMC-HVDC transmission system.Both theoretical and numerical studies have been undertaken to validate that the energy function proposed for hybrid AC/DC systems satisfies the conditions of an energy function.The work of this paper bridges the gap between the well-developed direct methods of transient stability analysis and power systems with MMC-HVDC transmission lines.
基金The research of the first author was supported by Hong Kong Baptist University,the Research Grants Council of Hong Kong.
文摘It is demonstrated that spectral methods can be used to improve the accuracy of numerical solutions obtained by some lower order methods.More precisely,we can use spectral methods to postprocess numerical solutions of initial value differential equations.After a few number of iterations(say 3 to 4),the errors can decrease to a few orders of magnitude less.The iteration uses the Gauss-Seidel type strategy,which gives an explicit way of postprocessing.Numerical examples for ODEs,Hamiltonian system and integral equations are provided.They all indicate that the spectral processing technique can be a very useful way in improving the accuracy of the numerical solutions.In particular,for a Hamiltonian system accuracy is only one of the issues;some other conservative properties are even more important for large time simulations.The spectral postprocessing with the coarse-mesh symplectic initial guess can not only produce high accurate approximations but can also save a significant amount of computational time over the standard symplectic schemes.
基金supported by PRIN 2015 Variational methods with applications to problems in mathematical physics and geometry
文摘In the last years much progress has been achieved in KAM theory concerning bifurcation of quasi-periodic solutions of Hamiltonian or reversible partial differential equations.We provide an overview of the state of the art in this field.
