In this paper, the genera]ised two-dimensiona] differentia] transform method （DTM） of solving the time-fractiona] coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The pr...In this paper, the genera]ised two-dimensiona] differentia] transform method （DTM） of solving the time-fractiona] coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial. An illustrative example shows that the genera]ised two-dimensional DTM is effective for the coupled equations.展开更多
By means of sn-function expansion method and cn-function expansion method, several kinds of explicit solutions to the coupled KdV equations with variable coefficients are obtained, which include three sets of periodic...By means of sn-function expansion method and cn-function expansion method, several kinds of explicit solutions to the coupled KdV equations with variable coefficients are obtained, which include three sets of periodic wave-like solutions. These solutions degenerate to solitary wave-like solutions at a certain limit. Some new solutions are presented.展开更多
The energy preserving average vector field （AVF） method is applied to the coupled Schr6dinger-KdV equations. Two energy preserving schemes are constructed by using Fourier pseudospectral method in space direction di...The energy preserving average vector field （AVF） method is applied to the coupled Schr6dinger-KdV equations. Two energy preserving schemes are constructed by using Fourier pseudospectral method in space direction discretization. In order to accelerate our simulation, the split-step technique is used. The numerical experiments show that the non-splitting scheme and splitting scheme are both effective, and have excellent long time numerical behavior. The comparisons show that the splitting scheme is faster than the non-splitting scheme, but it is not as good as the non-splitting scheme in preserving the invariants.展开更多
This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV）...This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV） equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.展开更多
Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the mod- ified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling ...Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the mod- ified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling wave solutions and white noise functional solutions are obtained, including the generalized exponential, hyperbolic, and trigonometric types.展开更多
In this paper, we consider the homotopy perturbation method (HPM) to obtain the exact solution of Hirota-Satsuma Coupled KdV equation. The results reveal that the proposed method is very effective and simple and can b...In this paper, we consider the homotopy perturbation method (HPM) to obtain the exact solution of Hirota-Satsuma Coupled KdV equation. The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear mathematical problems.展开更多
The purpose of this paper is to develop a hybridized discontinuous Galerkin(HDG)method for solving the Ito-type coupled KdV system.In fact,we use the HDG method for discre-tizing the space variable and the backward Eu...The purpose of this paper is to develop a hybridized discontinuous Galerkin(HDG)method for solving the Ito-type coupled KdV system.In fact,we use the HDG method for discre-tizing the space variable and the backward Euler explicit method for the time variable.To linearize the system,the time-lagging approach is also applied.The numerical stability of the method in the sense of the L2 norm is proved using the energy method under certain assumptions on the stabilization parameters for periodic or homogeneous Dirichlet bound-ary conditions.Numerical experiments confirm that the HDG method is capable of solving the system efficiently.It is observed that the best possible rate of convergence is achieved by the HDG method.Also,it is being illustrated numerically that the corresponding con-servation laws are satisfied for the approximate solutions of the Ito-type coupled KdV sys-tem.Thanks to the numerical experiments,it is verified that the HDG method could be more efficient than the LDG method for solving some Ito-type coupled KdV systems by comparing the corresponding computational costs and orders of convergence.展开更多
A special coupled KdV equation is proved to be the Painlevé property by the Kruskal's simplification ofWTC method.In order to search new exact solutions of the coupled KdV equation,Hirota's bilinear direc...A special coupled KdV equation is proved to be the Painlevé property by the Kruskal's simplification ofWTC method.In order to search new exact solutions of the coupled KdV equation,Hirota's bilinear direct method andthe conjugate complex number method of exponential functions are applied to this system.As a result,new analyticalcomplexiton and soliton solutions are obtained synchronously in a physical field.Then their structures,time evolutionand interaction properties are further discussed graphically.展开更多
In this paper, we present a multi-symplectic Hamiltonian formulation of the coupled Schrtidinger-KdV equations （CS＇KE） with periodic boundary conditions. Then we develop a novel multi-symplectic Fourier pseudospect...In this paper, we present a multi-symplectic Hamiltonian formulation of the coupled Schrtidinger-KdV equations （CS＇KE） with periodic boundary conditions. Then we develop a novel multi-symplectic Fourier pseudospectral （MSFP） scheme for the CSKE. In numerical experiments, we compare the MSFP method with the Crank-Nicholson （CN） method. Our results show high accuracy, effectiveness, and good ability of conserving the invariants of the MSFP method.展开更多
The present paper deals with the numerical solution of the coupled Schrodinger-KdV equations using the elementfree Galerkin （EFG） method which is based on the moving least-square approximation. Instead of traditiona...The present paper deals with the numerical solution of the coupled Schrodinger-KdV equations using the elementfree Galerkin （EFG） method which is based on the moving least-square approximation. Instead of traditional mesh oriented methods such as the finite difference method （FDM） and the finite element method （FEM）, this method needs only scattered nodes in the domain. For this scheme, a variational method is used to obtain discrete equations and the essential boundary conditions are enforced by the penalty method. In numerical experiments, the results are presented and compared with the findings of the finite element method, the radial basis functions method, and an analytical solution to confirm the good accuracy of the presented scheme.展开更多
The hybrid lattice, known as a discrete Korteweg-de Vries (KdV) equation, is found to be a discrete modified Korteweg-de Vries (mKdV) equation in this paper. The coupled hybrid lattice, which is pointed to be a discre...The hybrid lattice, known as a discrete Korteweg-de Vries (KdV) equation, is found to be a discrete modified Korteweg-de Vries (mKdV) equation in this paper. The coupled hybrid lattice, which is pointed to be a discrete coupled KdV system, is also found to be discrete form of a coupled mKdV systems. Delayed differential reduction system and pure difference systems are derived from the coupled hybrid system by means of the symmetry reduction approach. Cnoidal wave, positon and negaton solutions for the coupled hybrid system are proposed.展开更多
In this paper,two types of fractional nonlinear equations in Caputo sense,time-fractional Newell–Whitehead equation(FNWE)and time-fractional generalized Hirota–Satsuma coupled KdV system(HS-cKdVS),are investigated b...In this paper,two types of fractional nonlinear equations in Caputo sense,time-fractional Newell–Whitehead equation(FNWE)and time-fractional generalized Hirota–Satsuma coupled KdV system(HS-cKdVS),are investigated by means of the q-homotopy analysis method(q-HAM).The approximate solutions of the proposed equations are constructed in the form of a convergent series and are compared with the corresponding exact solutions.Due to the presence of the auxiliary parameter h in this method,just a few terms of the series solution are required in order to obtain better approximation.For the sake of visualization,the numerical results obtained in this paper are graphically displayed with the help of Maple.展开更多
基金Project supported by the Natural Science Foundation of Inner Mongolia of China (Grant No. 20080404MS0104)the Young Scientists Fund of Inner Mongolia University of China (Grant No. ND0811)
文摘In this paper, the genera]ised two-dimensiona] differentia] transform method （DTM） of solving the time-fractiona] coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial. An illustrative example shows that the genera]ised two-dimensional DTM is effective for the coupled equations.
文摘By means of sn-function expansion method and cn-function expansion method, several kinds of explicit solutions to the coupled KdV equations with variable coefficients are obtained, which include three sets of periodic wave-like solutions. These solutions degenerate to solitary wave-like solutions at a certain limit. Some new solutions are presented.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 90203001, 10475055, 90503006, and 10547124
The authors are indebted to Dr. F. Huang and Prof. Y. Chen for their helpful discussions.
基金The project supported by the National Fundamental Research Program of China(973 Program)under Grant No.2007CB814800National Natural Science Foundation of China under Grant No.10601028
基金supported by the National Natural Science Foundation of China(Grant No.91130013)the Open Foundation of State Key Laboratory of HighPerformance Computing of China
文摘The energy preserving average vector field （AVF） method is applied to the coupled Schr6dinger-KdV equations. Two energy preserving schemes are constructed by using Fourier pseudospectral method in space direction discretization. In order to accelerate our simulation, the split-step technique is used. The numerical experiments show that the non-splitting scheme and splitting scheme are both effective, and have excellent long time numerical behavior. The comparisons show that the splitting scheme is faster than the non-splitting scheme, but it is not as good as the non-splitting scheme in preserving the invariants.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10771019 and 10826107)
文摘This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV） equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.
文摘Variable coefficients and Wick-type stochastic fractional coupled KdV equations are investigated. By using the mod- ified fractional sub-equation method, Hermite transform, and white noise theory the exact travelling wave solutions and white noise functional solutions are obtained, including the generalized exponential, hyperbolic, and trigonometric types.
文摘In this paper, we consider the homotopy perturbation method (HPM) to obtain the exact solution of Hirota-Satsuma Coupled KdV equation. The results reveal that the proposed method is very effective and simple and can be applied to other nonlinear mathematical problems.
文摘The purpose of this paper is to develop a hybridized discontinuous Galerkin(HDG)method for solving the Ito-type coupled KdV system.In fact,we use the HDG method for discre-tizing the space variable and the backward Euler explicit method for the time variable.To linearize the system,the time-lagging approach is also applied.The numerical stability of the method in the sense of the L2 norm is proved using the energy method under certain assumptions on the stabilization parameters for periodic or homogeneous Dirichlet bound-ary conditions.Numerical experiments confirm that the HDG method is capable of solving the system efficiently.It is observed that the best possible rate of convergence is achieved by the HDG method.Also,it is being illustrated numerically that the corresponding con-servation laws are satisfied for the approximate solutions of the Ito-type coupled KdV sys-tem.Thanks to the numerical experiments,it is verified that the HDG method could be more efficient than the LDG method for solving some Ito-type coupled KdV systems by comparing the corresponding computational costs and orders of convergence.
文摘A special coupled KdV equation is proved to be the Painlevé property by the Kruskal's simplification ofWTC method.In order to search new exact solutions of the coupled KdV equation,Hirota's bilinear direct method andthe conjugate complex number method of exponential functions are applied to this system.As a result,new analyticalcomplexiton and soliton solutions are obtained synchronously in a physical field.Then their structures,time evolutionand interaction properties are further discussed graphically.
基金Project supported by the National Natural Science Foundation of China(Grant No.91130013)the Open Foundation of State Key Laboratory of High Performance Computing
文摘In this paper, we present a multi-symplectic Hamiltonian formulation of the coupled Schrtidinger-KdV equations （CS＇KE） with periodic boundary conditions. Then we develop a novel multi-symplectic Fourier pseudospectral （MSFP） scheme for the CSKE. In numerical experiments, we compare the MSFP method with the Crank-Nicholson （CN） method. Our results show high accuracy, effectiveness, and good ability of conserving the invariants of the MSFP method.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11072117 and 61074142)the Natural Science Foundation of Zhejiang Province,China(Grant No.Y6110007)+3 种基金Scientific Research Fund of Zhejiang Provincial Education Department,China(Grant No.Z201119278)the Natural Science Foundation of Ningbo City(Grant Nos.2012A610152 and 2012A610038)the Disciplinary Project of Ningbo City,China(Grant No.SZXL1067)K.C.Wong Magna Fund in Ningbo University
文摘The present paper deals with the numerical solution of the coupled Schrodinger-KdV equations using the elementfree Galerkin （EFG） method which is based on the moving least-square approximation. Instead of traditional mesh oriented methods such as the finite difference method （FDM） and the finite element method （FEM）, this method needs only scattered nodes in the domain. For this scheme, a variational method is used to obtain discrete equations and the essential boundary conditions are enforced by the penalty method. In numerical experiments, the results are presented and compared with the findings of the finite element method, the radial basis functions method, and an analytical solution to confirm the good accuracy of the presented scheme.
基金The project supported by National Natural Science Foundation of China under Grant No. 10071033 and the Natural Science Foundation of Jiangsu Province under Grant No. BK2002003. Acknowledgments 0ne of the authors （S.P. Qian） is indebted to Prof. S.Y. Lou for his helpful discussions.
基金＊Supported by the National Natural Science Foundation of China under Grant No. 60772023, by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. SKLSDE-07-001, Beijing University of Aeronautics and Astronautics, by the National Basic Research Program of China （973 Program） under Grant No. 2005CB321901, and by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos. 20060006024 and 200800130006, Chinese Ministry of Education.
基金Supported by the Natural Science Foundation of Guangdong Province of China under Grant No. 10452840301004616the National Natural Science Foundation of China under Grant No. 61001018the Scientific Research Foundation for the Doctors of University of Electronic Science and Technology of China Zhongshan Institute under Grant No. 408YKQ09
文摘The hybrid lattice, known as a discrete Korteweg-de Vries (KdV) equation, is found to be a discrete modified Korteweg-de Vries (mKdV) equation in this paper. The coupled hybrid lattice, which is pointed to be a discrete coupled KdV system, is also found to be discrete form of a coupled mKdV systems. Delayed differential reduction system and pure difference systems are derived from the coupled hybrid system by means of the symmetry reduction approach. Cnoidal wave, positon and negaton solutions for the coupled hybrid system are proposed.
基金supported by the National Natural Science Foundation of China(Grant No.12271433)。
文摘In this paper,two types of fractional nonlinear equations in Caputo sense,time-fractional Newell–Whitehead equation(FNWE)and time-fractional generalized Hirota–Satsuma coupled KdV system(HS-cKdVS),are investigated by means of the q-homotopy analysis method(q-HAM).The approximate solutions of the proposed equations are constructed in the form of a convergent series and are compared with the corresponding exact solutions.Due to the presence of the auxiliary parameter h in this method,just a few terms of the series solution are required in order to obtain better approximation.For the sake of visualization,the numerical results obtained in this paper are graphically displayed with the help of Maple.
