Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fer...Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fermi-Pasta-Ulam model. As a result, several types of similarity reductions are obtained. It is easy to show that the nonlinear wave equation is not integrable under the sense of AblowRz's conjecture from the reduction results obtained. In addition, kink-shaped solitary wave solutions, which are of important physical significance, are found for HMBEDT based on the obtained reduction equation.展开更多
In this paper, the travelling wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some e...In this paper, the travelling wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some exact explicit travelling solitary wave solutions for the above equation are obtained. As a result, some minor errors and some known results in the previousl literature are clarified and improved.展开更多
The compact implicit integration factor （cIIF） method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF me...The compact implicit integration factor （cIIF） method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger （NLS） equation and the complex Ginzburg-Landau （GL） equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.展开更多
The Jacobian elliptic function expansion method for nonlinear differential-different equations and its algorithm are presented by using some relations among ten Jacobian elliptic functions and successfully construct m...The Jacobian elliptic function expansion method for nonlinear differential-different equations and its algorithm are presented by using some relations among ten Jacobian elliptic functions and successfully construct more new exact doubly-periodic solutions of the integrable discrete nonlinear Schrodinger equation. When the modulous m → 1or 0, doubly-periodic solutions degenerate to solitonic solutions including bright soliton, dark soliton, new solitons as well as trigonometric function solutions.展开更多
The method of nonlinearization of spectral problems is developed to the defocusing nonlinear Schr(o|¨)dingerequation.As an application,an integrable decomposition of the defocusing nonlinear Schr(o|¨)dinger ...The method of nonlinearization of spectral problems is developed to the defocusing nonlinear Schr(o|¨)dingerequation.As an application,an integrable decomposition of the defocusing nonlinear Schr(o|¨)dinger equation is presented.展开更多
As a sequel to our recent work [1], in which a control framework was developed for large-scale joint swarms of unmanned ground (UGV) and aerial (UAV) vehicles, the present paper proposes cognitive and meta-cognitive s...As a sequel to our recent work [1], in which a control framework was developed for large-scale joint swarms of unmanned ground (UGV) and aerial (UAV) vehicles, the present paper proposes cognitive and meta-cognitive supervisor models for this kind of distributed robotic system. The cognitive supervisor model is a formalization of the recently Nobel-awarded research in brain science on mammalian and human path integration and navigation, performed by the hippocampus. This is formalized here as an adaptive Hamiltonian path integral, and efficiently simulated for implementation on robotic vehicles as a pair of coupled nonlinear Schr?dinger equations. The meta-cognitive supervisor model is a modal logic of actions and plans that hinges on a weak causality relation that specifies when atoms may change their values without specifying that they must change. This relatively simple logic is decidable yet sufficiently expressive to support the level of inference needed in our application. The atoms and action primitives of the logic framework also provide a straight-forward way of connecting the meta-cognitive supervisor with the cognitive supervisor, with other modules, and to the meta-cognitive supervisors of other robotic platforms in the swarm.展开更多
In this paper we propose stochastic multi-symplectic conservation law for stochastic Hamiltonian partial differential equations,and develop a stochastic multisymplectic method for numerically solving a kind of stochas...In this paper we propose stochastic multi-symplectic conservation law for stochastic Hamiltonian partial differential equations,and develop a stochastic multisymplectic method for numerically solving a kind of stochastic nonlinear Schrodinger equations.It is shown that the stochasticmulti-symplecticmethod preserves themultisymplectic structure,the discrete charge conservation law,and deduces the recurrence relation of the discrete energy.Numerical experiments are performed to verify the good behaviors of the stochastic multi-symplectic method in cases of both solitary wave and collision.展开更多
In this paper,we establish a family of symplectic integrators for a class of high order Schrodinger equations with trapped terms.First,we find its symplectic structure and reduce it to a finite dimensional Hamilton sy...In this paper,we establish a family of symplectic integrators for a class of high order Schrodinger equations with trapped terms.First,we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization.Then we apply the symplectic Euler method to the Hamiltonian system.It is demonstrated that the scheme not only preserves symplectic geometry structure of the original system,but also does not require to resolve coupled nonlinear algebraic equations which is different with the general implicit symplectic schemes.The linear stability of the symplectic Euler scheme and the errors of the numerical solutions are investigated.It shows that the semi-explicit scheme is conditionally stable,first order accurate in time and 2l th order accuracy in space.Numerical tests suggest that the symplectic integrators are more effective than non-symplectic ones,such as backward Euler integrators.展开更多
The dynamical self-trapping of an excitation propagating on one-dimensional of different sizes with nextnearest neighbor (NNN) interaction is studied by means of an explicit fourth order symplectic integrator. Using l...The dynamical self-trapping of an excitation propagating on one-dimensional of different sizes with nextnearest neighbor (NNN) interaction is studied by means of an explicit fourth order symplectic integrator. Using localized initial conditions, the time-averaged occupation probability of the initial site is investigated which is a function of the degree of nonlinearity and the linear coupling strengths. The self-trapping transition occurs at larger values of the nonlinearity parameter as the NNN coupling strength of the lattice increases for fixed size. Furthermore, given NNN coupling strength, the self-trapping properties for different sizes are considered which are some different from the case with general nearest neighbor (NN) interaction.展开更多
This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grü...This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference(WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schr?dinger(FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations(ODEs) in matrix formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson(CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor(c IIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ~2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.展开更多
文摘Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fermi-Pasta-Ulam model. As a result, several types of similarity reductions are obtained. It is easy to show that the nonlinear wave equation is not integrable under the sense of AblowRz's conjecture from the reduction results obtained. In addition, kink-shaped solitary wave solutions, which are of important physical significance, are found for HMBEDT based on the obtained reduction equation.
基金supported by the Research Foundation of Education Bureau of Hubei Province,China (Grant No Z200612001)the Natural Science Foundation of Yangtze University (Grant No 20061222)
文摘In this paper, the travelling wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some exact explicit travelling solitary wave solutions for the above equation are obtained. As a result, some minor errors and some known results in the previousl literature are clarified and improved.
文摘The compact implicit integration factor （cIIF） method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear SchrSdinger （NLS） equation and the complex Ginzburg-Landau （GL） equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.
文摘The Jacobian elliptic function expansion method for nonlinear differential-different equations and its algorithm are presented by using some relations among ten Jacobian elliptic functions and successfully construct more new exact doubly-periodic solutions of the integrable discrete nonlinear Schrodinger equation. When the modulous m → 1or 0, doubly-periodic solutions degenerate to solitonic solutions including bright soliton, dark soliton, new solitons as well as trigonometric function solutions.
基金Supported by the National Natural Science Foundation of China under Grant No.10871165
文摘The method of nonlinearization of spectral problems is developed to the defocusing nonlinear Schr(o|¨)dingerequation.As an application,an integrable decomposition of the defocusing nonlinear Schr(o|¨)dinger equation is presented.
文摘As a sequel to our recent work [1], in which a control framework was developed for large-scale joint swarms of unmanned ground (UGV) and aerial (UAV) vehicles, the present paper proposes cognitive and meta-cognitive supervisor models for this kind of distributed robotic system. The cognitive supervisor model is a formalization of the recently Nobel-awarded research in brain science on mammalian and human path integration and navigation, performed by the hippocampus. This is formalized here as an adaptive Hamiltonian path integral, and efficiently simulated for implementation on robotic vehicles as a pair of coupled nonlinear Schr?dinger equations. The meta-cognitive supervisor model is a modal logic of actions and plans that hinges on a weak causality relation that specifies when atoms may change their values without specifying that they must change. This relatively simple logic is decidable yet sufficiently expressive to support the level of inference needed in our application. The atoms and action primitives of the logic framework also provide a straight-forward way of connecting the meta-cognitive supervisor with the cognitive supervisor, with other modules, and to the meta-cognitive supervisors of other robotic platforms in the swarm.
基金supported by the NNSFC(No.11001009)supported by the Director Foundation of GUCAS,the NNSFC(No.11071251)supported by the Foundation of CAS and the NNSFC(No.11021101,No.91130003).
文摘In this paper we propose stochastic multi-symplectic conservation law for stochastic Hamiltonian partial differential equations,and develop a stochastic multisymplectic method for numerically solving a kind of stochastic nonlinear Schrodinger equations.It is shown that the stochasticmulti-symplecticmethod preserves themultisymplectic structure,the discrete charge conservation law,and deduces the recurrence relation of the discrete energy.Numerical experiments are performed to verify the good behaviors of the stochastic multi-symplectic method in cases of both solitary wave and collision.
基金supported by the Provincial Natural Science Foundation of Jiangxi(No.2008GQS0054)the Foundation of Department of Education Jiangxi province(No.GJJ09147)+1 种基金the Foundation of Jiangxi Normal University(Nos.2057 and 2390)State Key Laboratory of Scientific and Engineering Computing,CAS.This work is partially supported by the Provincial Natural Science Foundation of Anhui(No.090416227).
文摘In this paper,we establish a family of symplectic integrators for a class of high order Schrodinger equations with trapped terms.First,we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization.Then we apply the symplectic Euler method to the Hamiltonian system.It is demonstrated that the scheme not only preserves symplectic geometry structure of the original system,but also does not require to resolve coupled nonlinear algebraic equations which is different with the general implicit symplectic schemes.The linear stability of the symplectic Euler scheme and the errors of the numerical solutions are investigated.It shows that the semi-explicit scheme is conditionally stable,first order accurate in time and 2l th order accuracy in space.Numerical tests suggest that the symplectic integrators are more effective than non-symplectic ones,such as backward Euler integrators.
基金Supported by National Natural Science Foundation of China under Grant No.11271246Natural Science Research Project of Henan Education Department under Grant No.2011B110024+1 种基金Research Fund for Luoyang Normal University under Grant No.qnjj-2009-02for Henan Polytechnic University under Grant No.Q2012-30A
文摘The dynamical self-trapping of an excitation propagating on one-dimensional of different sizes with nextnearest neighbor (NNN) interaction is studied by means of an explicit fourth order symplectic integrator. Using localized initial conditions, the time-averaged occupation probability of the initial site is investigated which is a function of the degree of nonlinearity and the linear coupling strengths. The self-trapping transition occurs at larger values of the nonlinearity parameter as the NNN coupling strength of the lattice increases for fixed size. Furthermore, given NNN coupling strength, the self-trapping properties for different sizes are considered which are some different from the case with general nearest neighbor (NN) interaction.
基金supported by National Natural Science Foundation of China(Grant Nos.61573008 and 61703290)Laboratory of Computational Physics(Grant No.6142A0502020717)National Science Foundation of USA(Grant No.DMS-1620108)
文摘This paper proposes and analyzes an efficient finite difference scheme for the two-dimensional nonlinear Schr?dinger(NLS) equation involving fractional Laplacian. The scheme is based on a weighted and shifted Grünwald-Letnikov difference(WSGD) operator for the spatial fractional Laplacian. We prove that the proposed method preserves the mass and energy conservation laws in semi-discrete formulations. By introducing the differentiation matrices, the semi-discrete fractional nonlinear Schr?dinger(FNLS) equation can be rewritten as a system of nonlinear ordinary differential equations(ODEs) in matrix formulations. Two kinds of time discretization methods are proposed for the semi-discrete formulation. One is based on the Crank-Nicolson(CN) method which can be proved to preserve the fully discrete mass and energy conservation. The other one is the compact implicit integration factor(c IIF) method which demands much less computational effort. It can be shown that the cIIF scheme can approximate CN scheme with the error O(τ~2). Finally numerical results are presented to demonstrate the method’s conservation, accuracy, efficiency and the capability of capturing blow-up.
