The construction of moral education in primary and secondary schools is facingmany dilemmas,such as formalism,passivity,task-type,Tacitus trap and new media interaction.There is a high degree of coupling between the p...The construction of moral education in primary and secondary schools is facingmany dilemmas,such as formalism,passivity,task-type,Tacitus trap and new media interaction.There is a high degree of coupling between the progressive,stepped and interlocking supply of Problem Chain mode and the demand of moral education construction in primary and secondary schools.Problem Chain mode is helpful to solve the problems of teaching content,teaching effects and educational object in moral education construction in primary and secondary schools.展开更多
The present work deals with the calculation of transition probability between two diabatic potentials coupled by any arbitrary coupling. The method presented in this work is applicable to any type of coupling while fo...The present work deals with the calculation of transition probability between two diabatic potentials coupled by any arbitrary coupling. The method presented in this work is applicable to any type of coupling while for numerical calculations we have assumed the arbitrary coupling as Gaussian coupling. This arbitrary coupling is expressed as a collection of Dirac delta functions and by the use of the transfer matrix technique the transition probability from one diabatic potential to another diabatic potential is calculated. We examine our approach by considering the case of two constant potentials coupled by Gaussian coupling as an arbitrary coupling.展开更多
Significant progress has been made in mixed boundary-value problems associated with three-dimensional（3D） crack and contact analyses of advanced materials featuring more complexities compared to the conventional iso...Significant progress has been made in mixed boundary-value problems associated with three-dimensional（3D） crack and contact analyses of advanced materials featuring more complexities compared to the conventional isotropic elastic materials.These include material anisotropy and multifield coupling,two typical characteristics of most current multifunctional materials.In this paper we try to present a state-of-the-art description of 3D exact/analytical solutions derived for crack and contact problems of elastic solids with both transverse isotropy and multifield coupling in the latest decade by the potential theory method in the spirit of V.I.Fabrikant.whose ingenious breakthrough brings new vigor and vitality to the old research subject of classical potential theory.We are particularly interested in crack and contact problems with certain nonlinear features.Emphasis is also placed on the coupling between the temperature field（or the like） and other physical fields（e.g.,elastic,electric,and magnetic fields）.We further highlight the practical significance of 3D contact solutions,in particular in applications related to modern scanning probe microscopes.展开更多
A new isospectral problem is firstly presented, then we derive integrable system of soliton hierarchy. Also we obtain new integrable couplings of the generalized Kaup-Newell soliton equations hierarchy and its Hamilto...A new isospectral problem is firstly presented, then we derive integrable system of soliton hierarchy. Also we obtain new integrable couplings of the generalized Kaup-Newell soliton equations hierarchy and its Hamiltonian structures by using Tu scheme and the quadratic-form identity. The method can be generalized to other soliton hierarchy.展开更多
This paper derives new discrete integrable system based on discrete isospectral problem. It shows that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, int...This paper derives new discrete integrable system based on discrete isospectral problem. It shows that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrable couplings of the obtained system is given by means of semi-direct sums of Lie algebras.展开更多
This paper reports a multiscale analysis method to predict the thermomechanical coupling performance of composite structures with quasi-periodic properties.In these material structures,the configurations are periodic,...This paper reports a multiscale analysis method to predict the thermomechanical coupling performance of composite structures with quasi-periodic properties.In these material structures,the configurations are periodic,and the material coefficients are quasi-periodic,i.e.,they depend not only on the microscale information but also on the macro location.Also,a mutual interaction between displacement and temperature fields is considered in the problem,which is our particular interest in this study.The multiscale asymptotic expansions of the temperature and displacement fields are constructed and associated error estimation in nearly pointwise sense is presented.Then,a finite element-difference algorithm based on the multiscale analysis method is brought forward in detail.Finally,some numerical examples are given.And the numerical results show that the multiscale method presented in this paper is effective and reliable to study the nonlinear thermo-mechanical coupling problem of composite structures with quasiperiodic properties.展开更多
This paper presents an overview of the recent progress of potential theory method in the analysis of mixed boundary value problems mainly stemming from three-dimensional crack or contact problems of multi-field couple...This paper presents an overview of the recent progress of potential theory method in the analysis of mixed boundary value problems mainly stemming from three-dimensional crack or contact problems of multi-field coupled media. This method was used to derive a series of exact three dimensional solutions which should be of great theoretical significance because most of them usually cannot be derived by other methods such as the transform method and the trial-and-error method. Further, many solutions are obtained in terms of elementary functions that enable us to treat more complicated problems easily. It is pointed out here that the method is usually only applicable to media characterizing transverse isotropy, from which, however, the results for the isotropic case can be readily obtained.展开更多
We propose a multi-field implicit finite element method for analyzing the electromechanical behavior of dielectric elastomers. This method is based on a four-field variational principle, which includes displacement an...We propose a multi-field implicit finite element method for analyzing the electromechanical behavior of dielectric elastomers. This method is based on a four-field variational principle, which includes displacement and electric potential for the electromechanical coupling analysis, and additional independent fields to address the incompressible constraint of the hyperelastic material. Linearization of the variational form and finite element discretization are adopted for the numerical implementation. A general FEM program framework is devel- oped using C＋＋ based on the open-source finite element library deal.II to implement this proposed algorithm. Numerical examples demonstrate the accuracy, convergence properties, mesh-independence properties, and scalability of this method. We also use the method for eigenvalue analysis of a dielectric elastomer actuator subject to electromechanical loadings. Our finite element implementation is available as an online supplementary material.展开更多
In the fracture problems of hydrophilic elastic materials under coupling effects of heat conduction, moisture diffusion and mechanical deformation, the conventional J-integral is no longer path independent. The value ...In the fracture problems of hydrophilic elastic materials under coupling effects of heat conduction, moisture diffusion and mechanical deformation, the conventional J-integral is no longer path independent. The value of J is unequal to the energy release rate in hygrothermal coupling cases. In the present paper, we derived a general form of the energy release rate for hygrothermal fracture problems of the hydrophilic elastic materials on the basis of energy balance equation in cracked areas. By introducing the constitutive relations and the essential equations of irreversible thermodynamics, a specific expression of the energy release rate was obtained, and the expression can be reformmulated as path independent integrals, which is equivalent to the energy release rate of the fracture body. The path independence of the integrals is then verified numerically.展开更多
We present an efficient three-dimensional coupled-mode model based on the Fourier synthesis technique. In principle, this model is a one-way model, and hence provides satisfactory accuracy for problems where the forwa...We present an efficient three-dimensional coupled-mode model based on the Fourier synthesis technique. In principle, this model is a one-way model, and hence provides satisfactory accuracy for problems where the forward scattering dominates. At the same time, this model provides an efficiency gain of an order of magnitude or more over two-way coupled-mode models. This model can be applied to three-dimensional range-dependent problems with a slowly varying bathymetry or internal waves. A numerical example of the latter is demonstrated in this work. Comparisons of both accuracy and efficiency between the present model and a benchmark model are also provided.展开更多
A generalized AKNS isospectral problem where the trace of corresponding spectral matrix is not zero, is transformed to a new isospectral problem where the trace of the resulting matrix is zero, by using transformation...A generalized AKNS isospectral problem where the trace of corresponding spectral matrix is not zero, is transformed to a new isospectral problem where the trace of the resulting matrix is zero, by using transformation of Lax pairs, and these two spectral problems lead to the same hierarchy of equations. The authors started from the transformed spectral problem and constructed a new loop algebra which has not appeared before, and obtained the integrable coupling of the generalized AKNS hierarchy. Specially, the integrable couplings of the KdV equation and MKdV equation are obtained.展开更多
This paper introduces a new stabilized finite element method for the coupled Stokes and Darcy problem based on the nonconforming Crouzeix-Raviart element. Optimal error estimates for the fluid velocity and pressure ar...This paper introduces a new stabilized finite element method for the coupled Stokes and Darcy problem based on the nonconforming Crouzeix-Raviart element. Optimal error estimates for the fluid velocity and pressure are derived. A numerical example is presented to verify the theoretical predictions.展开更多
Liouville integrable discrete integrable system is derived based on discrete isospectral problem. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. ...Liouville integrable discrete integrable system is derived based on discrete isospectral problem. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrable couplings of the obtained system is given by means of semi-direct sums of Lie algebras.展开更多
Two types of Lie algebras are presented,from which two integrable couplings associated with the Tuisospectral problem are obtained,respectively.One of them possesses the Hamiltonian structure generated by a linearisom...Two types of Lie algebras are presented,from which two integrable couplings associated with the Tuisospectral problem are obtained,respectively.One of them possesses the Hamiltonian structure generated by a linearisomorphism and the quadratic-form identity.An approach for working out the double integrable couplings of the sameintegrable system is presented in the paper.展开更多
In this study we describe an FEM-based methodology to solve the coupled fluid-structure problem due to squeeze film effects present in vibratory MEMS devices, such as resonators, gyroscopes, and acoustic transducers. ...In this study we describe an FEM-based methodology to solve the coupled fluid-structure problem due to squeeze film effects present in vibratory MEMS devices, such as resonators, gyroscopes, and acoustic transducers. The aforementioned devices often consist of a plate-like structure that vibrates normal to a fixed substrate, and is generally not perfectly vacuum packed. This results in a thin film of air being sandwiched between the moving plate and the fixed substrate, which behaves like a squeeze film offering both stiffness and damping. Typically, such structures are actuated electro-statically, necessitating the thin air gap for improving the efficiency of actuation and the sensitivity of detection. To accurately model these devices the squeeze film effect must be incorporated. Extensive literature is present on mod- eling squeeze film effects for rigid motion for both perforated as well as non-perforated plates. Studies which model the plate elasticity often use approximate mode shapes as input to the 2D Reynolds Equation. Recent works which try to solve the coupled fluid elasticity problem, report iterative FEM-based solution strategies for the 2D Reynolds Equation coupled with the 3D elasticity Equation. In this work we present a FEM-based single step solution for the coupled problem at hand, using only one type of element (27 node 3D brick). The structure is modeled with 27 node brick elements of which the lowest layer of nodes is also treated as the fluid domain (2D) and the integrals over fluid domain are evaluated for these nodes only. We also apply an electrostatic loading to our model by considering an equivalent electro-static pressure load on the top surface of the structure. Thus we solve the coupled 2D-fluid-3D-structure problem in a single step, using only one element type. The FEM results show good agreement with both existing analytical solutions and published experimental data.展开更多
The presence of non-gray radiative properties in a reheating furnace’s medium that absorbs,emits,and involves non-gray creates more complex radiative heat transfer problems.Furthermore,it adds difficulty to solving t...The presence of non-gray radiative properties in a reheating furnace’s medium that absorbs,emits,and involves non-gray creates more complex radiative heat transfer problems.Furthermore,it adds difficulty to solving the coupled conduction,convection,and radiation problem,leading to suboptimal efficiency that fails to meet real-time control demands.To overcome this difficulty,comparable gray radiative properties of non-gray media are proposed and estimated by solving an inverse problem.However,the required iteration numbers by using a least-squares method are too many and resulted in a very low inverse efficiency.It is necessary to present an efficient method for the equivalence.The Levenberg-Marquardt algorithm is utilized to solve the inverse problem of coupled heat transfer,and the gray-equivalent radiative characteristics are successfully recovered.It is our intention that the issue of low inverse efficiency,which has been observed when the least-squares method is employed,will be resolved.To enhance the performance of the Levenberg-Marquardt algorithm,a modification is implemented for determining the damping factor.Detailed investigations are also conducted to evaluate its accuracy,stability of convergence,efficiency,and robustness of the algorithm.Subsequently,a comparison is made between the results achieved using each method.展开更多
By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville...By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.展开更多
We establish the existence of positive solutions for singular boundary value problems of coupled systems? The proof relies on Schauder’s fixed point theorem. Some recent results in the literature are generalized and ...We establish the existence of positive solutions for singular boundary value problems of coupled systems? The proof relies on Schauder’s fixed point theorem. Some recent results in the literature are generalized and improved.展开更多
By using cone expansion-compression theorem in this paper, we study boundary value problems for a coupled system of nonlinear third-order differential equation. Some sufficient conditions are obtained which guarantee ...By using cone expansion-compression theorem in this paper, we study boundary value problems for a coupled system of nonlinear third-order differential equation. Some sufficient conditions are obtained which guarantee the boundary value problems for a coupled system of nonlinear third-order differential equation has at least one positive solution. Some examples are given to verify our results.展开更多
We study the coupled mKdV equation by the dressing method via local Riemann-Hilbert problem. With the help of the Lax pairs, we obtain the matrix Riemann-Hilbert problem with zeros. The explicit solutions for the coup...We study the coupled mKdV equation by the dressing method via local Riemann-Hilbert problem. With the help of the Lax pairs, we obtain the matrix Riemann-Hilbert problem with zeros. The explicit solutions for the coupled mKdV equation are derived with the aid of the regularization of the Riemann-Hilbert problem.展开更多
文摘The construction of moral education in primary and secondary schools is facingmany dilemmas,such as formalism,passivity,task-type,Tacitus trap and new media interaction.There is a high degree of coupling between the progressive,stepped and interlocking supply of Problem Chain mode and the demand of moral education construction in primary and secondary schools.Problem Chain mode is helpful to solve the problems of teaching content,teaching effects and educational object in moral education construction in primary and secondary schools.
文摘The present work deals with the calculation of transition probability between two diabatic potentials coupled by any arbitrary coupling. The method presented in this work is applicable to any type of coupling while for numerical calculations we have assumed the arbitrary coupling as Gaussian coupling. This arbitrary coupling is expressed as a collection of Dirac delta functions and by the use of the transfer matrix technique the transition probability from one diabatic potential to another diabatic potential is calculated. We examine our approach by considering the case of two constant potentials coupled by Gaussian coupling as an arbitrary coupling.
基金supported by the National Natural Science Foundation of China(Grant 11321202)the Specialized Research Fund for the Doctoral Program of Higher Education(Grant 20130101110120)
文摘Significant progress has been made in mixed boundary-value problems associated with three-dimensional（3D） crack and contact analyses of advanced materials featuring more complexities compared to the conventional isotropic elastic materials.These include material anisotropy and multifield coupling,two typical characteristics of most current multifunctional materials.In this paper we try to present a state-of-the-art description of 3D exact/analytical solutions derived for crack and contact problems of elastic solids with both transverse isotropy and multifield coupling in the latest decade by the potential theory method in the spirit of V.I.Fabrikant.whose ingenious breakthrough brings new vigor and vitality to the old research subject of classical potential theory.We are particularly interested in crack and contact problems with certain nonlinear features.Emphasis is also placed on the coupling between the temperature field（or the like） and other physical fields（e.g.,elastic,electric,and magnetic fields）.We further highlight the practical significance of 3D contact solutions,in particular in applications related to modern scanning probe microscopes.
基金Supported by the Natural Science Foundation of China under Grant Nos. 61072147, 11071159, and 10971031by the Natural Science Foundation of Shanghai and Zhejiang Province under Grant Nos. 09ZR1410800 and Y6100791+1 种基金the Shanghai Shuguang Tracking Project under Grant No. 08GG01the Shanghai Leading Academic Discipline Project under Grant No. J50101
文摘A new isospectral problem is firstly presented, then we derive integrable system of soliton hierarchy. Also we obtain new integrable couplings of the generalized Kaup-Newell soliton equations hierarchy and its Hamiltonian structures by using Tu scheme and the quadratic-form identity. The method can be generalized to other soliton hierarchy.
文摘This paper derives new discrete integrable system based on discrete isospectral problem. It shows that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrable couplings of the obtained system is given by means of semi-direct sums of Lie algebras.
基金financially supported by the National Natural Science Foundation of China(11501449)the Fundamental Research Funds for the Central Universities(3102017zy043)+2 种基金the China Postdoctoral Science Foundation(2016T91019)the fund of the State Key Laboratory of Solidification Processing in NWPU(SKLSP201628)the Scientific Research Program Funded by Shaanxi Provincial Education Department(14JK1353).
文摘This paper reports a multiscale analysis method to predict the thermomechanical coupling performance of composite structures with quasi-periodic properties.In these material structures,the configurations are periodic,and the material coefficients are quasi-periodic,i.e.,they depend not only on the microscale information but also on the macro location.Also,a mutual interaction between displacement and temperature fields is considered in the problem,which is our particular interest in this study.The multiscale asymptotic expansions of the temperature and displacement fields are constructed and associated error estimation in nearly pointwise sense is presented.Then,a finite element-difference algorithm based on the multiscale analysis method is brought forward in detail.Finally,some numerical examples are given.And the numerical results show that the multiscale method presented in this paper is effective and reliable to study the nonlinear thermo-mechanical coupling problem of composite structures with quasiperiodic properties.
基金Project (No. 10372088) supported by the National Natural Science Foundation of China
文摘This paper presents an overview of the recent progress of potential theory method in the analysis of mixed boundary value problems mainly stemming from three-dimensional crack or contact problems of multi-field coupled media. This method was used to derive a series of exact three dimensional solutions which should be of great theoretical significance because most of them usually cannot be derived by other methods such as the transform method and the trial-and-error method. Further, many solutions are obtained in terms of elementary functions that enable us to treat more complicated problems easily. It is pointed out here that the method is usually only applicable to media characterizing transverse isotropy, from which, however, the results for the isotropic case can be readily obtained.
基金the support under A*STAR SERC grant (132-183-0025)
文摘We propose a multi-field implicit finite element method for analyzing the electromechanical behavior of dielectric elastomers. This method is based on a four-field variational principle, which includes displacement and electric potential for the electromechanical coupling analysis, and additional independent fields to address the incompressible constraint of the hyperelastic material. Linearization of the variational form and finite element discretization are adopted for the numerical implementation. A general FEM program framework is devel- oped using C＋＋ based on the open-source finite element library deal.II to implement this proposed algorithm. Numerical examples demonstrate the accuracy, convergence properties, mesh-independence properties, and scalability of this method. We also use the method for eigenvalue analysis of a dielectric elastomer actuator subject to electromechanical loadings. Our finite element implementation is available as an online supplementary material.
基金The project supported by the Key Project of Chinese Ministry of Education (03145)the Science Fund of Southwest Jiaotong University
文摘In the fracture problems of hydrophilic elastic materials under coupling effects of heat conduction, moisture diffusion and mechanical deformation, the conventional J-integral is no longer path independent. The value of J is unequal to the energy release rate in hygrothermal coupling cases. In the present paper, we derived a general form of the energy release rate for hygrothermal fracture problems of the hydrophilic elastic materials on the basis of energy balance equation in cracked areas. By introducing the constitutive relations and the essential equations of irreversible thermodynamics, a specific expression of the energy release rate was obtained, and the expression can be reformmulated as path independent integrals, which is equivalent to the energy release rate of the fracture body. The path independence of the integrals is then verified numerically.
基金Supported by the National Natural Science Foundation of China under Grant No 11774374the Natural Science Foundation of Shandong Province of China under Grant No ZR2016AL10
文摘We present an efficient three-dimensional coupled-mode model based on the Fourier synthesis technique. In principle, this model is a one-way model, and hence provides satisfactory accuracy for problems where the forward scattering dominates. At the same time, this model provides an efficiency gain of an order of magnitude or more over two-way coupled-mode models. This model can be applied to three-dimensional range-dependent problems with a slowly varying bathymetry or internal waves. A numerical example of the latter is demonstrated in this work. Comparisons of both accuracy and efficiency between the present model and a benchmark model are also provided.
文摘A generalized AKNS isospectral problem where the trace of corresponding spectral matrix is not zero, is transformed to a new isospectral problem where the trace of the resulting matrix is zero, by using transformation of Lax pairs, and these two spectral problems lead to the same hierarchy of equations. The authors started from the transformed spectral problem and constructed a new loop algebra which has not appeared before, and obtained the integrable coupling of the generalized AKNS hierarchy. Specially, the integrable couplings of the KdV equation and MKdV equation are obtained.
基金Project supported by the Science and Technology Foundation of Sichuan Province(No. 05GG006-006-2)
文摘This paper introduces a new stabilized finite element method for the coupled Stokes and Darcy problem based on the nonconforming Crouzeix-Raviart element. Optimal error estimates for the fluid velocity and pressure are derived. A numerical example is presented to verify the theoretical predictions.
文摘Liouville integrable discrete integrable system is derived based on discrete isospectral problem. It is shown that the hierarchy is completely integrable in the Liouville sense and possesses bi-Hamiltonian structure. Finally, integrable couplings of the obtained system is given by means of semi-direct sums of Lie algebras.
基金National Natural Science Foundation of China under Grant No.10471139
文摘Two types of Lie algebras are presented,from which two integrable couplings associated with the Tuisospectral problem are obtained,respectively.One of them possesses the Hamiltonian structure generated by a linearisomorphism and the quadratic-form identity.An approach for working out the double integrable couplings of the sameintegrable system is presented in the paper.
文摘In this study we describe an FEM-based methodology to solve the coupled fluid-structure problem due to squeeze film effects present in vibratory MEMS devices, such as resonators, gyroscopes, and acoustic transducers. The aforementioned devices often consist of a plate-like structure that vibrates normal to a fixed substrate, and is generally not perfectly vacuum packed. This results in a thin film of air being sandwiched between the moving plate and the fixed substrate, which behaves like a squeeze film offering both stiffness and damping. Typically, such structures are actuated electro-statically, necessitating the thin air gap for improving the efficiency of actuation and the sensitivity of detection. To accurately model these devices the squeeze film effect must be incorporated. Extensive literature is present on mod- eling squeeze film effects for rigid motion for both perforated as well as non-perforated plates. Studies which model the plate elasticity often use approximate mode shapes as input to the 2D Reynolds Equation. Recent works which try to solve the coupled fluid elasticity problem, report iterative FEM-based solution strategies for the 2D Reynolds Equation coupled with the 3D elasticity Equation. In this work we present a FEM-based single step solution for the coupled problem at hand, using only one type of element (27 node 3D brick). The structure is modeled with 27 node brick elements of which the lowest layer of nodes is also treated as the fluid domain (2D) and the integrals over fluid domain are evaluated for these nodes only. We also apply an electrostatic loading to our model by considering an equivalent electro-static pressure load on the top surface of the structure. Thus we solve the coupled 2D-fluid-3D-structure problem in a single step, using only one element type. The FEM results show good agreement with both existing analytical solutions and published experimental data.
基金supported by the Na⁃tional Natural Science Foundation of China(No.12172078)the Fundamental Research Funds for the Central Univer⁃sities(No.DUT24MS007).
文摘The presence of non-gray radiative properties in a reheating furnace’s medium that absorbs,emits,and involves non-gray creates more complex radiative heat transfer problems.Furthermore,it adds difficulty to solving the coupled conduction,convection,and radiation problem,leading to suboptimal efficiency that fails to meet real-time control demands.To overcome this difficulty,comparable gray radiative properties of non-gray media are proposed and estimated by solving an inverse problem.However,the required iteration numbers by using a least-squares method are too many and resulted in a very low inverse efficiency.It is necessary to present an efficient method for the equivalence.The Levenberg-Marquardt algorithm is utilized to solve the inverse problem of coupled heat transfer,and the gray-equivalent radiative characteristics are successfully recovered.It is our intention that the issue of low inverse efficiency,which has been observed when the least-squares method is employed,will be resolved.To enhance the performance of the Levenberg-Marquardt algorithm,a modification is implemented for determining the damping factor.Detailed investigations are also conducted to evaluate its accuracy,stability of convergence,efficiency,and robustness of the algorithm.Subsequently,a comparison is made between the results achieved using each method.
基金National Natural Science Foundation of China under Grant No.60572113the Natural Science Foundation of Shandong Province of China under Grant No.Q2006A04the Talents Foundation of Taishan College under Grant No.Y05-2-01
文摘By considering a new discrete isospectral eigenvalue problem, a hierarchy of lattice soliton equations of rational type are derived. It is shown that each equation in the resulting hierarchy is integrable in Liouville sense and possessing bi-Hamiltonian structure. Two types of semi-direct sums of Lie algebras are proposed, by using of which a practicable way to construct discrete integrable couplings is introduced. As applications, two kinds of discrete integrable couplings of the resulting system are worked out.
文摘We establish the existence of positive solutions for singular boundary value problems of coupled systems? The proof relies on Schauder’s fixed point theorem. Some recent results in the literature are generalized and improved.
基金Foundation item： Supported by the National Natural Science Foundation of China（10801001） Supported by the Natural Science Foundation of Anhui Province（1208085MA13）
文摘By using cone expansion-compression theorem in this paper, we study boundary value problems for a coupled system of nonlinear third-order differential equation. Some sufficient conditions are obtained which guarantee the boundary value problems for a coupled system of nonlinear third-order differential equation has at least one positive solution. Some examples are given to verify our results.
文摘We study the coupled mKdV equation by the dressing method via local Riemann-Hilbert problem. With the help of the Lax pairs, we obtain the matrix Riemann-Hilbert problem with zeros. The explicit solutions for the coupled mKdV equation are derived with the aid of the regularization of the Riemann-Hilbert problem.
