The relation between the toal variation of classical field theory and the multisymplectic structure is shown. Then the multisymplectic structure and the corresponding multisymplectic conservation of the coupled nonlin...The relation between the toal variation of classical field theory and the multisymplectic structure is shown. Then the multisymplectic structure and the corresponding multisymplectic conservation of the coupled nonlinear Schroedinger system are obtained directly from the variational principle.展开更多
The multisymplectic geometry for the seismic wave equation is presented in this paper.The local energy conservation law,the local momentum evolution equations,and the multisymplectic form are derived directly from the...The multisymplectic geometry for the seismic wave equation is presented in this paper.The local energy conservation law,the local momentum evolution equations,and the multisymplectic form are derived directly from the variational principle.Based on the covariant Legendre transform,the multisymplectic Hamiltonian formulation is developed.Multisymplectic discretization and numerical experiments are also explored.展开更多
In the previous papers I and II, we have studied the difference discrete variational principle and the Euler?Lagrange cohomology in the framework of multi-parameter differential approach. We have gotten the difference...In the previous papers I and II, we have studied the difference discrete variational principle and the Euler?Lagrange cohomology in the framework of multi-parameter differential approach. We have gotten the difference discrete Euler?Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler?Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the Lagrangian and Hamiltonian formalisms. In this paper, we apply the difference discrete variational principle and Euler?Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms. We will show that either Hamiltonian schemes or Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler?Lagrange cohomological conditions are satisfied.展开更多
In this second paper of a series of papers, we explore the difference discrete versions for the Euler?Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in...In this second paper of a series of papers, we explore the difference discrete versions for the Euler?Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in both the Lagrangian and Hamiltonian formalisms for discrete mechanics and field theory in the framework of multi-parameter differential approach. In terms of the difference discrete Euler?Lagrange cohomological concepts, we show that the symplectic or multisymplectic geometry and their difference discrete structure-preserving properties can always be established not only in the solution spaces of the discrete Euler?Lagrange or canonical equations derived by the difference discrete variational principle but also in the function space in each case if and only if the relevant closed Euler?Lagrange cohomological conditions are satisfied.展开更多
We propose multisymplectic implicit and explicit Fourier pseudospectral methods for the Klein-Gordon-Schrodinger equations.We prove that the implicit method satisfies the charge conservation law exactly.Both methods p...We propose multisymplectic implicit and explicit Fourier pseudospectral methods for the Klein-Gordon-Schrodinger equations.We prove that the implicit method satisfies the charge conservation law exactly.Both methods provide accurate solutions in long-time computations and simulate the soliton collision well.The numerical results show the abilities of the two methods in preserving the charge,energy,and momentum conservation laws.展开更多
We explore the multisymplectic Fourier pseudospectral discretizations for the (3+1)-dimensional Klein-Gordon equation in this paper.The corresponding multisymplectic conservation laws are derived.Two kinds of explicit...We explore the multisymplectic Fourier pseudospectral discretizations for the (3+1)-dimensional Klein-Gordon equation in this paper.The corresponding multisymplectic conservation laws are derived.Two kinds of explicitsymplectic integrators in time are also presented.展开更多
We derive a new multisymplectic integrator for the Kawahara-type equation which is a fully explicit scheme and thus needs less computation cost. Multisympecticity of such scheme guarantees the long-time numerical beha...We derive a new multisymplectic integrator for the Kawahara-type equation which is a fully explicit scheme and thus needs less computation cost. Multisympecticity of such scheme guarantees the long-time numerical behaviors. Nu- merical experiments are presented to verify the accuracy of this scheme as well as the excellent performance on invariant preservation for three kinds of Kawahara-type equations.展开更多
In this paper, taking the 2+1-dimensional sine-Gordon equation as an example, we present the concatenating method to construct the multisymplectic schemes. The method is to discretizee independently the PDEs in differ...In this paper, taking the 2+1-dimensional sine-Gordon equation as an example, we present the concatenating method to construct the multisymplectic schemes. The method is to discretizee independently the PDEs in different directions with symplectic schemes, so that the multisymplectic schemes can be constructed by concatenating those symplectic schemes. By this method, we can construct multisymplectic schemes, including some widely used schemes with an accuracy of any order. The numerical simulation on the collisions of solitons are also proposed to illustrate the efficiency of the multisymplectic schemes.展开更多
In this paper, the multisymplectic Fourier pseudospectral scheme for initial-boundary value problems of nonlinear SchrSdinger equations with wave operator is considered. We investigate the local and global conservatio...In this paper, the multisymplectic Fourier pseudospectral scheme for initial-boundary value problems of nonlinear SchrSdinger equations with wave operator is considered. We investigate the local and global conservation properties of the multisymplectic discretization based on Fourier pseudospectral approximations. The local and global spatial conservation of energy is proved. The error estimates of local energy conservation law are also derived. Numerical experiments are presented to verify the theoretical predications.展开更多
A multisymplectic Fourier pseudo-spectral scheme, which exactly preserves the discrete multisym- plectic conservation law, is presented to solve the Klein-Gordon-SchrSdinger equations. The scheme is of spectral accura...A multisymplectic Fourier pseudo-spectral scheme, which exactly preserves the discrete multisym- plectic conservation law, is presented to solve the Klein-Gordon-SchrSdinger equations. The scheme is of spectral accuracy in space and of second order in time. The scheme preserves the discrete multisymplectic conservation law and the charge conservation law. Moreover, the residuals of some other conservation laws are derived for the geometric numerical integrator. Extensive numerical simulations illustrate the numerical behavior of the multisymplectic scheme, and demonstrate the correctness of the theoretical analysis.展开更多
In this paper, we introduce the multisymplecticstructure of the nonlinear wave equation, and prove that theclassical five-point scheme for the equation is multisymplec-tic. Numerical simulations of this multisymplecti...In this paper, we introduce the multisymplecticstructure of the nonlinear wave equation, and prove that theclassical five-point scheme for the equation is multisymplec-tic. Numerical simulations of this multisymplectic scheme onhighly oscillatory waves of the nonlinear Klein-Gordonequation and the collisions between kink and anti-kink soli-tons of the sine-Gordon equation are also provided. The mul-tisymplectic schemes do not need to discrete PDEs in thespace first as the symplectic schemes do and preserve notonly the geometric structure of the PDEs accurately, but alsotheir first integrals approximately such as the energy, themomentum and so on. Thus the multisymplectic schemeshave better numerical stability and long-time numerical be-havior than the energy-conserving scheme and the symplec-tic scheme.展开更多
The multisymplectic structure of the nonlinear wave equation is derived directly from the variational principle. In the numerical aspect, we present a multisymplectic nine points scheme which is equivalent to the mult...The multisymplectic structure of the nonlinear wave equation is derived directly from the variational principle. In the numerical aspect, we present a multisymplectic nine points scheme which is equivalent to the multisymplectic Preissman scheme. A series of numerical results are reported to illustrate the effectiveness of the scheme.展开更多
In this paper, a novel multisymplectic scheme is proposed for the coupled nonlinear Schrodinger-KdV （CNLS-KdV） equations. The CNLS-KdV equations are rewritten into the multisymplectic Hamiltonian form by introducing...In this paper, a novel multisymplectic scheme is proposed for the coupled nonlinear Schrodinger-KdV （CNLS-KdV） equations. The CNLS-KdV equations are rewritten into the multisymplectic Hamiltonian form by introducing some canonical momenta. To simulate the problem efficiently, the CNLS-KdV equations are approximated by a high order compact method in space which preserves N semi-discrete multisymplectic conservation laws. We then discretize the semi-discrete system by using a symplectic midpoint scheme in time. Thus, a full-discrete multisymplectic scheme is obtained for the CNLS-KdV equations. The conservation laws of the full-discrete scheme are analyzed. Some numerical experiments are presented to further verify the convergence and conservation laws of the new scheme.展开更多
A composition method for constructing high order multisymplectic integrators is presented in this paper. The basic idea is to apply composition method to both the time and the space directions. We also obtain a genera...A composition method for constructing high order multisymplectic integrators is presented in this paper. The basic idea is to apply composition method to both the time and the space directions. We also obtain a general formula for composition method.展开更多
In view of the complex structure and environment,the dynamic analysis on deoxyribonucleic acid(DNA)is a challenge in the biophysics field.Considering the local interaction with ribonucleic acid(RNA)-polymerase as well...In view of the complex structure and environment,the dynamic analysis on deoxyribonucleic acid(DNA)is a challenge in the biophysics field.Considering the local interaction with ribonucleic acid(RNA)-polymerase as well as the dissipative effect of cellular fluid,a coupling sine-Gordon-type dynamic model is used to describe the rotational motions of the bases in DNA.First,the approximate symmetric form is constructed.Then,the wave form and the wave velocity of the kink solution to the proposed dynamic model are investigated by a Runge-Kutta structure-preserving scheme based on the generalized multi-symplectic idea.The numerical results indicate that,the strengthening of the local interaction between DNA and RNA-polymerase described by the coupling potential makes the form of the kink solution steep,while the appearance of the friction between DNA and cellular fluid makes the form of the kink solution flat.In addition,the appearance of the friction decreases the velocities of both the symplectic configuration and the anti-symplectic configuration with different degrees.The above findings are beneficial to comprehend the DNA transcription mechanism.展开更多
Solvent-free nanofluids hold promise for many technologically significant applications.The liquid-like behavior,a typical rheological property of solvent-free nanofluids,has aroused considerable interests.However,ther...Solvent-free nanofluids hold promise for many technologically significant applications.The liquid-like behavior,a typical rheological property of solvent-free nanofluids,has aroused considerable interests.However,there has been still lack of efficient methods to predict and control the liquid-like behavior of solvent-free nanofluids.In this paper,we propose a semi-discrete dynamic system with stochastic excitation describing the temperature change effects on the rheological property of multiwall carbon nanotubes(MWCNTs)modified by grafting sulfonic acid terminated organosilanes as corona and tertiary amine as canopy,which is a typical covalent-type solvent-free nanofluid system.The vibration of the grafting branches is simulated by employing a structure-preserving approach,and the shear force of grafting branches at the fixed end is computed subsequently.By taking the shear forces as an excitation acting on the MWCNTs,the axial motion of the MWCNTs is solved with the 7-point Gauss-Kronrod quadrature rule.The critical temperature associated with the appearance of the liquid-like behavior as well as the upper bound of the moving speed of the modified MWCNTs is determined,which can be used to predict and control the liquid-like behavior of the modified MWCNTs in engineering applications.展开更多
In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM ⊕∧nT*M for an m-dimensional manifold. As an application, we revisit Nambu-Poisson ...In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM ⊕∧nT*M for an m-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an (n + 1)-vector field π is closed under the higher-order Dorfman bracket iff π is a Nambu-Poisson structure. Consequently, there is an induced Leibniz algebroid structure on ∧nT*M. The graph of an (n+1)-form ω is closed under the higher-order Dorfman bracket iff ω is a premultisymplectic structure of order n, i.e., dω = 0. Furthermore, there is a Lie algebroid structure on the admissible bundle A ∧nT*M. In particular, for a 2-plectic structure, it induces the Lie 2-algebra structure given in (Baez, Hoffnung and Rogers, 2010).展开更多
In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi equation.The two equations are in the family of integrable peakon equations,and both have very rich geometric properties.Based on these geome...In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi equation.The two equations are in the family of integrable peakon equations,and both have very rich geometric properties.Based on these geometric structures,we construct the geometric numerical integrators for simulating their soliton solutions.The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties,however they also have the significant difference,for example there exist the shock wave solutions for the Degasperis-Procesi equation.By using the symplectic Fourier pseudo-spectral integrator,we simulate the peakon solutions of the two equations.To illustrate the smooth solitons and shock wave solutions of the DP equation,we use the splitting technique and combine the composition methods.In the numerical experiments,comparisons of these two kinds of methods are presented in terms of accuracy,computational cost and invariants preservation.展开更多
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10401033 and 10471145 and the Key Project of Knowledge Innovation of CAS under Grant No. KZCX1-SW-18
文摘The relation between the toal variation of classical field theory and the multisymplectic structure is shown. Then the multisymplectic structure and the corresponding multisymplectic conservation of the coupled nonlinear Schroedinger system are obtained directly from the variational principle.
文摘The multisymplectic geometry for the seismic wave equation is presented in this paper.The local energy conservation law,the local momentum evolution equations,and the multisymplectic form are derived directly from the variational principle.Based on the covariant Legendre transform,the multisymplectic Hamiltonian formulation is developed.Multisymplectic discretization and numerical experiments are also explored.
文摘In the previous papers I and II, we have studied the difference discrete variational principle and the Euler?Lagrange cohomology in the framework of multi-parameter differential approach. We have gotten the difference discrete Euler?Lagrange equations and canonical ones for the difference discrete versions of classical mechanics and field theory as well as the difference discrete versions for the Euler?Lagrange cohomology and applied them to get the necessary and sufficient condition for the symplectic or multisymplectic geometry preserving properties in both the Lagrangian and Hamiltonian formalisms. In this paper, we apply the difference discrete variational principle and Euler?Lagrange cohomological approach directly to the symplectic and multisymplectic algorithms. We will show that either Hamiltonian schemes or Lagrangian ones in both the symplectic and multisymplectic algorithms are variational integrators and their difference discrete symplectic structure-preserving properties can always be established not only in the solution space but also in the function space if and only if the related closed Euler?Lagrange cohomological conditions are satisfied.
文摘In this second paper of a series of papers, we explore the difference discrete versions for the Euler?Lagrange cohomology and apply them to the symplectic or multisymplectic geometry and their preserving properties in both the Lagrangian and Hamiltonian formalisms for discrete mechanics and field theory in the framework of multi-parameter differential approach. In terms of the difference discrete Euler?Lagrange cohomological concepts, we show that the symplectic or multisymplectic geometry and their difference discrete structure-preserving properties can always be established not only in the solution spaces of the discrete Euler?Lagrange or canonical equations derived by the difference discrete variational principle but also in the function space in each case if and only if the relevant closed Euler?Lagrange cohomological conditions are satisfied.
基金Project supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No. 11201169)the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 10KJB110001)
文摘We propose multisymplectic implicit and explicit Fourier pseudospectral methods for the Klein-Gordon-Schrodinger equations.We prove that the implicit method satisfies the charge conservation law exactly.Both methods provide accurate solutions in long-time computations and simulate the soliton collision well.The numerical results show the abilities of the two methods in preserving the charge,energy,and momentum conservation laws.
基金supported by National Natural Science Foundation of China under Grant No.40774069partially by the National Hi-Tech Research and Development Program of China under Crant No.2006AA09A102-08State Key Basic Research Program under Grant No.2007CB209603
文摘We explore the multisymplectic Fourier pseudospectral discretizations for the (3+1)-dimensional Klein-Gordon equation in this paper.The corresponding multisymplectic conservation laws are derived.Two kinds of explicitsymplectic integrators in time are also presented.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11271195 and 11271196)the Project of Graduate Education Innovation of Jiangsu Province,China(Grant No.CXZZ12-0385)
文摘We derive a new multisymplectic integrator for the Kawahara-type equation which is a fully explicit scheme and thus needs less computation cost. Multisympecticity of such scheme guarantees the long-time numerical behaviors. Nu- merical experiments are presented to verify the accuracy of this scheme as well as the excellent performance on invariant preservation for three kinds of Kawahara-type equations.
基金This work was supported by the National Natural Science Foundation of China Innovation Group(No.40221503)the CAS Hundred Talent Project,the National Key Development Planning Project for the Basic Research(No.1999032081)the National Natural Science Foundation of China(Grant No.10226012).
文摘In this paper, taking the 2+1-dimensional sine-Gordon equation as an example, we present the concatenating method to construct the multisymplectic schemes. The method is to discretizee independently the PDEs in different directions with symplectic schemes, so that the multisymplectic schemes can be constructed by concatenating those symplectic schemes. By this method, we can construct multisymplectic schemes, including some widely used schemes with an accuracy of any order. The numerical simulation on the collisions of solitons are also proposed to illustrate the efficiency of the multisymplectic schemes.
基金This work was supported by E-Institutes of Shanghai Municlpal Education Commission N.E03004.
文摘In this paper, the multisymplectic Fourier pseudospectral scheme for initial-boundary value problems of nonlinear SchrSdinger equations with wave operator is considered. We investigate the local and global conservation properties of the multisymplectic discretization based on Fourier pseudospectral approximations. The local and global spatial conservation of energy is proved. The error estimates of local energy conservation law are also derived. Numerical experiments are presented to verify the theoretical predications.
基金supported by National Natural Science Foundation of China(Grant Nos.10901074,11271171,91130003,11001009 and 11101399)the Province Natural Science Foundation of Jiangxi(Grant No. 20114BAB201011)+2 种基金the Foundation of Department of Education of Jiangxi Province(Grant No.GJJ12174)the State Key Laboratory of Scientific and Engineering Computing,CASsupported by the Youth Growing Foundation of Jiangxi Normal University in 2010
文摘A multisymplectic Fourier pseudo-spectral scheme, which exactly preserves the discrete multisym- plectic conservation law, is presented to solve the Klein-Gordon-SchrSdinger equations. The scheme is of spectral accuracy in space and of second order in time. The scheme preserves the discrete multisymplectic conservation law and the charge conservation law. Moreover, the residuals of some other conservation laws are derived for the geometric numerical integrator. Extensive numerical simulations illustrate the numerical behavior of the multisymplectic scheme, and demonstrate the correctness of the theoretical analysis.
文摘In this paper, we introduce the multisymplecticstructure of the nonlinear wave equation, and prove that theclassical five-point scheme for the equation is multisymplec-tic. Numerical simulations of this multisymplectic scheme onhighly oscillatory waves of the nonlinear Klein-Gordonequation and the collisions between kink and anti-kink soli-tons of the sine-Gordon equation are also provided. The mul-tisymplectic schemes do not need to discrete PDEs in thespace first as the symplectic schemes do and preserve notonly the geometric structure of the PDEs accurately, but alsotheir first integrals approximately such as the energy, themomentum and so on. Thus the multisymplectic schemeshave better numerical stability and long-time numerical be-havior than the energy-conserving scheme and the symplec-tic scheme.
基金Supported by the Special Funds for Major State Basic Research Projects G.032800the National Key Lab.Project G.40023001.
文摘The multisymplectic structure of the nonlinear wave equation is derived directly from the variational principle. In the numerical aspect, we present a multisymplectic nine points scheme which is equivalent to the multisymplectic Preissman scheme. A series of numerical results are reported to illustrate the effectiveness of the scheme.
基金This work is supported by the NNSFC （Nos. 11771213, 41504078, 11301234, 11271171）, the National Key Research and Development Project of China （No. 2016YFC0600310）, the Major Projects of Natural Sciences of University in Jiangsu Province of China （No. 15KJA110002） and the Priority Academic Program Development of Jiangsu Higher Education Institutions, the Provincial Natural Science Foundation of Jiangxi （Nos. 20161ACB20006, 20142BCB23009, 20151BAB 201012）.
文摘In this paper, a novel multisymplectic scheme is proposed for the coupled nonlinear Schrodinger-KdV （CNLS-KdV） equations. The CNLS-KdV equations are rewritten into the multisymplectic Hamiltonian form by introducing some canonical momenta. To simulate the problem efficiently, the CNLS-KdV equations are approximated by a high order compact method in space which preserves N semi-discrete multisymplectic conservation laws. We then discretize the semi-discrete system by using a symplectic midpoint scheme in time. Thus, a full-discrete multisymplectic scheme is obtained for the CNLS-KdV equations. The conservation laws of the full-discrete scheme are analyzed. Some numerical experiments are presented to further verify the convergence and conservation laws of the new scheme.
基金This work is subsidized by the special funds for major state basic research projects (No.1999032800).
文摘A composition method for constructing high order multisymplectic integrators is presented in this paper. The basic idea is to apply composition method to both the time and the space directions. We also obtain a general formula for composition method.
基金the National Natural Science Foundation of China(Nos.11972284 and11672241)the Fund for Distinguished Young Scholars of Shaanxi Province of China(No.2019JC-29)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment of China(No.GZ19103)。
文摘In view of the complex structure and environment,the dynamic analysis on deoxyribonucleic acid(DNA)is a challenge in the biophysics field.Considering the local interaction with ribonucleic acid(RNA)-polymerase as well as the dissipative effect of cellular fluid,a coupling sine-Gordon-type dynamic model is used to describe the rotational motions of the bases in DNA.First,the approximate symmetric form is constructed.Then,the wave form and the wave velocity of the kink solution to the proposed dynamic model are investigated by a Runge-Kutta structure-preserving scheme based on the generalized multi-symplectic idea.The numerical results indicate that,the strengthening of the local interaction between DNA and RNA-polymerase described by the coupling potential makes the form of the kink solution steep,while the appearance of the friction between DNA and cellular fluid makes the form of the kink solution flat.In addition,the appearance of the friction decreases the velocities of both the symplectic configuration and the anti-symplectic configuration with different degrees.The above findings are beneficial to comprehend the DNA transcription mechanism.
基金supported by the National Natural Science Foundation of China(Nos.12172281 and 11972284)the Distinguished Young Scholars of Shaanxi Province of China(No.2019JC-29)+2 种基金the Foundation Strengthening Programme Technical Area Fund of Shaanxi Province of China(No.2021-JCJQ-JJ-0565)the Science and Technology Innovation Team of Shaanxi Province of China(No.2022TD-61)the Youth Innovation Team of Shaanxi Universities and Doctoral Dissertation Innovation Fund of Xi’an University of Technology of China(Nos.252072016 and 252072115)。
文摘Solvent-free nanofluids hold promise for many technologically significant applications.The liquid-like behavior,a typical rheological property of solvent-free nanofluids,has aroused considerable interests.However,there has been still lack of efficient methods to predict and control the liquid-like behavior of solvent-free nanofluids.In this paper,we propose a semi-discrete dynamic system with stochastic excitation describing the temperature change effects on the rheological property of multiwall carbon nanotubes(MWCNTs)modified by grafting sulfonic acid terminated organosilanes as corona and tertiary amine as canopy,which is a typical covalent-type solvent-free nanofluid system.The vibration of the grafting branches is simulated by employing a structure-preserving approach,and the shear force of grafting branches at the fixed end is computed subsequently.By taking the shear forces as an excitation acting on the MWCNTs,the axial motion of the MWCNTs is solved with the 7-point Gauss-Kronrod quadrature rule.The critical temperature associated with the appearance of the liquid-like behavior as well as the upper bound of the moving speed of the modified MWCNTs is determined,which can be used to predict and control the liquid-like behavior of the modified MWCNTs in engineering applications.
基金supported by National Natural Science Foundation of China(Grant No. 10871007)US-China CMR Noncommutative Geometry (Grant No. 10911120391/A0109)+1 种基金China Postdoctoral Science Foundation (Grant No. 20090451267)Science Research Foundation for Excellent Young Teachers of Mathematics School at Jilin University
文摘In this paper, we study the algebraic properties of the higher analogues of Courant algebroid structures on the direct sum bundle TM ⊕∧nT*M for an m-dimensional manifold. As an application, we revisit Nambu-Poisson structures and multisymplectic structures. We prove that the graph of an (n + 1)-vector field π is closed under the higher-order Dorfman bracket iff π is a Nambu-Poisson structure. Consequently, there is an induced Leibniz algebroid structure on ∧nT*M. The graph of an (n+1)-form ω is closed under the higher-order Dorfman bracket iff ω is a premultisymplectic structure of order n, i.e., dω = 0. Furthermore, there is a Lie algebroid structure on the admissible bundle A ∧nT*M. In particular, for a 2-plectic structure, it induces the Lie 2-algebra structure given in (Baez, Hoffnung and Rogers, 2010).
基金This research was supported by the National Natural Science Foundation of China 11271357,11271195 and 41504078by the CSC,the Foundation for Innovative Research Groups of the NNSFC 11321061 and the ITER-China Program 2014GB124005。
文摘In this paper,we study the Camassa-Holm equation and the Degasperis-Procesi equation.The two equations are in the family of integrable peakon equations,and both have very rich geometric properties.Based on these geometric structures,we construct the geometric numerical integrators for simulating their soliton solutions.The Camassa-Holm equation and the Degasperis-Procesi equation have many common properties,however they also have the significant difference,for example there exist the shock wave solutions for the Degasperis-Procesi equation.By using the symplectic Fourier pseudo-spectral integrator,we simulate the peakon solutions of the two equations.To illustrate the smooth solitons and shock wave solutions of the DP equation,we use the splitting technique and combine the composition methods.In the numerical experiments,comparisons of these two kinds of methods are presented in terms of accuracy,computational cost and invariants preservation.
