Two types of symmetry of a generalized Zakharov-Kuznetsov equation are obtained via a direct symmetry method. By selecting suitable parameters occurring in the symmetries, we also find some symmetry reductions and new...Two types of symmetry of a generalized Zakharov-Kuznetsov equation are obtained via a direct symmetry method. By selecting suitable parameters occurring in the symmetries, we also find some symmetry reductions and new explicit solutions of the generalized Zakharov-Kuznetsov equation.展开更多
In this paper, we study a differential-difference equation associated with discrete 3 × 3 matrix spectral problem. Based on gauge transformation of the spectral problm, Darboux transformation of the differential-...In this paper, we study a differential-difference equation associated with discrete 3 × 3 matrix spectral problem. Based on gauge transformation of the spectral problm, Darboux transformation of the differential-difference equation is given. In order to solve the differential-difference equation, a systematic algebraic algorithm is given. As an application, explicit soliton solutions of the differential-difference equation are given.展开更多
An integrable （2＋1）-dimensional coupled mKdV equation is decomposed into two （1 ＋1）-dimensional soliton systems, which is produced from the compatible condition of three spectral problems. With the help of decom...An integrable （2＋1）-dimensional coupled mKdV equation is decomposed into two （1 ＋1）-dimensional soliton systems, which is produced from the compatible condition of three spectral problems. With the help of decomposition and the Darboux transformation of two （1＋1）-dimensional soliton systems, some interesting explicit solutions of these soliton equations are obtained.展开更多
Daxboux transformation with multi-parameters for the Boussinesq-Burgers （B-B） equation is derived. For an application, some important explicit solutions of the B-B equation are obtained, including 2N-soliton solutio...Daxboux transformation with multi-parameters for the Boussinesq-Burgers （B-B） equation is derived. For an application, some important explicit solutions of the B-B equation are obtained, including 2N-soliton solution and periodic solution. Finally, some elegant and interesting figures are plotted.展开更多
The modified Korteweg-de Vries （mKdV） typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With th...The modified Korteweg-de Vries （mKdV） typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With the aid of symbolic computation, the discrete matrix spectral problem for that system is constructed. Darboux transformation for that system is established based on the resulting spectral problem. Explicit solutions are derived via the Darboux transformation. Structures of those solutions are shown graphically, which might be helpful to understand some physical processes in fluids, plasmas, and optics.展开更多
A generalized Drinfel＇d Sokolov-Wilson （DSW） equation and its Lax pair are proposed. A Daorboux transformation for the generalized DSW equation is constructed with the help of the gauge transformation between spect...A generalized Drinfel＇d Sokolov-Wilson （DSW） equation and its Lax pair are proposed. A Daorboux transformation for the generalized DSW equation is constructed with the help of the gauge transformation between spectral problems, from which a Darboux transformation for the DSW equation is obtained through a reduction technique. As an application of the Darboux transformations, we give some explicit solutions of the generalized DSW equation and DEW equation such as rational solutions, soliton solutions, periodic solutions.展开更多
his paper studies the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation using the (G'/G)-expand method, we obtain many new explicit solutions of the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equ...his paper studies the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation using the (G'/G)-expand method, we obtain many new explicit solutions of the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation, which include hyperbolic function solutions, trigonometric function solutions and rational function solutions and so on.展开更多
In this work,a method is put forward to obtain the dynamic solution efficiently and accurately for a large-scale train-track-substructure(TTS)system.It is called implicit-explicit integration and multi-time-step solut...In this work,a method is put forward to obtain the dynamic solution efficiently and accurately for a large-scale train-track-substructure(TTS)system.It is called implicit-explicit integration and multi-time-step solution method(abbreviated as mI-nE-MTS method).The TTS system is divided into train-track subsystem and substruc-ture subsystem.Considering that the root cause of low effi-ciency of obtaining TTS solution lies in solving the alge-braic equation of the substructures,the high-efficient Zhai method,an explicit integration scheme,can be introduced to avoid matrix inversion process.The train-track system is solved by implicitly Park method.Moreover,it is known that the requirement of time step size differs for different sub-systems,integration methods and structural frequency response characteristics.A multi-time-step solution is pro-posed,in which time step size for the train-track subsystem and the substructure subsystem can be arbitrarily chosen once satisfying stability and precision demand,namely the time spent for m implicit integral steps is equal to n explicit integral steps,i.e.,mI=nE as mentioned above.The numeri-cal examples show the accuracy,efficiency,and engineering practicality of the proposed method.展开更多
The nonlocal symmetry of the mKdV equation is obtained from the known Lax pair; it is successfully localized to Lie point symmetries in the enlarged space by introducing suitable auxiliary dependent variables. For the...The nonlocal symmetry of the mKdV equation is obtained from the known Lax pair; it is successfully localized to Lie point symmetries in the enlarged space by introducing suitable auxiliary dependent variables. For the closed prolongation of the nonlocal symmetry, the details of the construction for a one-dimensional optimal system are presented. Furthermore, using the associated vector fields of the obtained symmetry, we give the reductions by the one-dimensional sub-algebras and the explicit analytic interaction solutions between cnoidal waves and kink solitary waves, which provide a way to study the interactions among these types of ocean waves. For some of the interesting solutions, the figures are given to show their properties.展开更多
In this paper by using tensor analysis we give the explicit expressions of the solution of the initial-value problem of homogeneous linear differential equations with constant coefficients and the nth-order homogeneou...In this paper by using tensor analysis we give the explicit expressions of the solution of the initial-value problem of homogeneous linear differential equations with constant coefficients and the nth-order homogeneous linear differential equation with constant coefficients. In fact, we give the general formula for calculating the elements of the matrix exp[At] . We also give the results when the characteristic equation has the repeated roots. The present method is simpler and better than the other methods.展开更多
This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-（3＋1）- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmet...This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-（3＋1）- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmetry algebra, symmetry group and group-invariant solutions are analysed. Otherwise, some exact explicit solutions are obtained from the corresponding （2＋1）-dimensional equation, the inviscid barotropic nondivergent vorticy equation. To show the properties and characters of these solutions, some plots as well as their possible physical meanings of the atmospheric circulation are given out.展开更多
Abstract： At first one of g-inverses of A （×） In＋Im（×） BT is given out, then the explicit solution to matrix equation AX ＋ XB = C is gained by using the method of matrix decomposition, finally, a nume...Abstract： At first one of g-inverses of A （×） In＋Im（×） BT is given out, then the explicit solution to matrix equation AX ＋ XB = C is gained by using the method of matrix decomposition, finally, a numerical example is obtained.展开更多
By using the compatibility method, many explicit solutions of the (1+1)-dimensional variable-coefficientBroer-Kaup system are constructed, which include new solutions expressed by error function, Bessel function, expo...By using the compatibility method, many explicit solutions of the (1+1)-dimensional variable-coefficientBroer-Kaup system are constructed, which include new solutions expressed by error function, Bessel function, exponentialfunction, and Airy function.Some figures of the solutions are given by the symbolic computation system Maple.展开更多
Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamilto...Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differentiM equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable ease of the two-dimensional Hamiltonian system.展开更多
Using the sign-invariant theory, we study the nonlinear reaction-diffusion systems. We also obtain some new explicit solutions to the nonlinear resulting systems.
Abstract By applying the Lie group method, the （2＋l）-dimensional soliton equation is reduced to some （1＋1）-dimensional nonlinear equations. Based upon some new explicit solutions of the （2＋1）-dimensional brea...Abstract By applying the Lie group method, the （2＋l）-dimensional soliton equation is reduced to some （1＋1）-dimensional nonlinear equations. Based upon some new explicit solutions of the （2＋1）-dimensional breaking soliton equation are obtained.展开更多
An integrable （2＋1）-dimensional Toda lattice with two discrete variables is investigated again, which is produced from a compatible condition of the Lax triad. The Darboux transformation for its spectral problems i...An integrable （2＋1）-dimensional Toda lattice with two discrete variables is investigated again, which is produced from a compatible condition of the Lax triad. The Darboux transformation for its spectral problems is found. As an application, explicit solutions of the （2＋1）-dimensional Toda equation with two discrete variables are obtained.展开更多
Nonlocal symmetry and explicit solution of the integrable Alice-Bob modified Korteweg-de Vries(ABm Kd V) equation is discussed, which has been established by the aid of the shifted parity and delayed time reversal to ...Nonlocal symmetry and explicit solution of the integrable Alice-Bob modified Korteweg-de Vries(ABm Kd V) equation is discussed, which has been established by the aid of the shifted parity and delayed time reversal to describe two-place events. Based on the Lax pair which contains the two-order partial derivative, the Lie symmetry group method is successfully applied to find the exact invariant solution for the AB-m Kd V equation with nonlocal symmetry by introducing one suitable auxiliary variable. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to some specific functions are derived. Figures show the physical phenomenon, that is, "the shifted parity and delayed time reversal to describe two-place events".展开更多
The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the first integral meth...The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the first integral method, in the form of trigonometric and exponential functions. The results show the first integral method is an efficient way to solve the coupled nonlinear equations and get rich explicit analytical solutions.展开更多
基金The project supported by Natural Science Foundation of Shandong Province of China under Grant 2004 zx 16The authors would like to thank professor Bai Cheng-Lin and the referees for their valuable advices.
文摘Two types of symmetry of a generalized Zakharov-Kuznetsov equation are obtained via a direct symmetry method. By selecting suitable parameters occurring in the symmetries, we also find some symmetry reductions and new explicit solutions of the generalized Zakharov-Kuznetsov equation.
基金Project supported by the Talent Foundation of the Northwest Sci-Tech University of Agriculture and Forestry (01140407)
文摘In this paper, we study a differential-difference equation associated with discrete 3 × 3 matrix spectral problem. Based on gauge transformation of the spectral problm, Darboux transformation of the differential-difference equation is given. In order to solve the differential-difference equation, a systematic algebraic algorithm is given. As an application, explicit soliton solutions of the differential-difference equation are given.
基金the Special Funds for Major State Basic Research Project of China under No.G2000077301
文摘An integrable （2＋1）-dimensional coupled mKdV equation is decomposed into two （1 ＋1）-dimensional soliton systems, which is produced from the compatible condition of three spectral problems. With the help of decomposition and the Darboux transformation of two （1＋1）-dimensional soliton systems, some interesting explicit solutions of these soliton equations are obtained.
文摘Daxboux transformation with multi-parameters for the Boussinesq-Burgers （B-B） equation is derived. For an application, some important explicit solutions of the B-B equation are obtained, including 2N-soliton solution and periodic solution. Finally, some elegant and interesting figures are plotted.
基金Supported by the National Natural Science Foundation of China under Grant No. 60772023by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. BUAA-SKLSDE-09KF-04+2 种基金Beijing University of Aeronautics and Astronautics, by the National Basic Research Program of China (973 Program) under Grant No. 2005CB321901the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos. 20060006024 and 200800130006Chinese Ministry of Education, and Scientific Research Common Program of Beijing Municipal Commission of Education under Grant No. KM201010772020
文摘The modified Korteweg-de Vries （mKdV） typed equations can be used to describe certain nonlinear phenomena in fluids, plasmas, and optics. In this paper, the discretized mKdV lattice equation is investigated. With the aid of symbolic computation, the discrete matrix spectral problem for that system is constructed. Darboux transformation for that system is established based on the resulting spectral problem. Explicit solutions are derived via the Darboux transformation. Structures of those solutions are shown graphically, which might be helpful to understand some physical processes in fluids, plasmas, and optics.
基金Supported by National Natural Science Foundation of China under Grant No.10871182Innovation Scientists and Technicians Troop Construction Projects of Henan Province
文摘A generalized Drinfel＇d Sokolov-Wilson （DSW） equation and its Lax pair are proposed. A Daorboux transformation for the generalized DSW equation is constructed with the help of the gauge transformation between spectral problems, from which a Darboux transformation for the DSW equation is obtained through a reduction technique. As an application of the Darboux transformations, we give some explicit solutions of the generalized DSW equation and DEW equation such as rational solutions, soliton solutions, periodic solutions.
文摘his paper studies the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation using the (G'/G)-expand method, we obtain many new explicit solutions of the generalized (2 + 1)-dimensional Zakharov-Kuznetsov equation, which include hyperbolic function solutions, trigonometric function solutions and rational function solutions and so on.
基金This work was supported by the National Natural Science Foundation of China(Grant Nos.52008404,U1934217 and 11790283)Science and Technology Research and Development Program Project of China Railway Group Limited(Major Special Project,No.2020-Special-02)the National Natural Science Foundation of Hunan Province(Grant No.2021JJ30850).
文摘In this work,a method is put forward to obtain the dynamic solution efficiently and accurately for a large-scale train-track-substructure(TTS)system.It is called implicit-explicit integration and multi-time-step solution method(abbreviated as mI-nE-MTS method).The TTS system is divided into train-track subsystem and substruc-ture subsystem.Considering that the root cause of low effi-ciency of obtaining TTS solution lies in solving the alge-braic equation of the substructures,the high-efficient Zhai method,an explicit integration scheme,can be introduced to avoid matrix inversion process.The train-track system is solved by implicitly Park method.Moreover,it is known that the requirement of time step size differs for different sub-systems,integration methods and structural frequency response characteristics.A multi-time-step solution is pro-posed,in which time step size for the train-track subsystem and the substructure subsystem can be arbitrarily chosen once satisfying stability and precision demand,namely the time spent for m implicit integral steps is equal to n explicit integral steps,i.e.,mI=nE as mentioned above.The numeri-cal examples show the accuracy,efficiency,and engineering practicality of the proposed method.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11075055 and 11275072)the Innovative Research Team Program of the National Natural Science Foundation of China(Grant No.61021004)+1 种基金the National High Technology Research and Development Program of China(Grant No.2011AA010101)the Shanghai Knowledge Service Platform for Trustworthy Internet of Things,China(Grant No.ZF1213)
文摘The nonlocal symmetry of the mKdV equation is obtained from the known Lax pair; it is successfully localized to Lie point symmetries in the enlarged space by introducing suitable auxiliary dependent variables. For the closed prolongation of the nonlocal symmetry, the details of the construction for a one-dimensional optimal system are presented. Furthermore, using the associated vector fields of the obtained symmetry, we give the reductions by the one-dimensional sub-algebras and the explicit analytic interaction solutions between cnoidal waves and kink solitary waves, which provide a way to study the interactions among these types of ocean waves. For some of the interesting solutions, the figures are given to show their properties.
文摘In this paper by using tensor analysis we give the explicit expressions of the solution of the initial-value problem of homogeneous linear differential equations with constant coefficients and the nth-order homogeneous linear differential equation with constant coefficients. In fact, we give the general formula for calculating the elements of the matrix exp[At] . We also give the results when the characteristic equation has the repeated roots. The present method is simpler and better than the other methods.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10735030,90718041 and 40975038)Shanghai Leading Academic Discipline Project(Grant No.B412)Program for Changjiang Scholars and Innovative Research Team in University(Grant No.IRT0734)
文摘This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-（3＋1）- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmetry algebra, symmetry group and group-invariant solutions are analysed. Otherwise, some exact explicit solutions are obtained from the corresponding （2＋1）-dimensional equation, the inviscid barotropic nondivergent vorticy equation. To show the properties and characters of these solutions, some plots as well as their possible physical meanings of the atmospheric circulation are given out.
文摘Abstract： At first one of g-inverses of A （×） In＋Im（×） BT is given out, then the explicit solution to matrix equation AX ＋ XB = C is gained by using the method of matrix decomposition, finally, a numerical example is obtained.
基金Supported by the Youth Nature Science Foundation of Anhui University of Technology in China under Grant No.QZ200823the Natural Science Foundation of the Anhui Higher Education Institutions of China under Grant No.KJ2010A043
文摘By using the compatibility method, many explicit solutions of the (1+1)-dimensional variable-coefficientBroer-Kaup system are constructed, which include new solutions expressed by error function, Bessel function, exponentialfunction, and Airy function.Some figures of the solutions are given by the symbolic computation system Maple.
基金the Funds for Basic Research Project under Grant Nos.06XJC033 and 2008Bl10003
文摘Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differentiM equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable ease of the two-dimensional Hamiltonian system.
基金National Natural Science Foundation of China under Grant Nos.10447007 and 10671156the Natural Science Foundation of Shaanxi Province of China under Grant No.2005A13
文摘Using the sign-invariant theory, we study the nonlinear reaction-diffusion systems. We also obtain some new explicit solutions to the nonlinear resulting systems.
基金The project supported by the Natural Science Foundation of Shandong Province of China under Grant No. 2004zx16
文摘Abstract By applying the Lie group method, the （2＋l）-dimensional soliton equation is reduced to some （1＋1）-dimensional nonlinear equations. Based upon some new explicit solutions of the （2＋1）-dimensional breaking soliton equation are obtained.
基金The project supported by the Special Funds for Major State Basic Research Project under Grant No. G2000077301
文摘An integrable （2＋1）-dimensional Toda lattice with two discrete variables is investigated again, which is produced from a compatible condition of the Lax triad. The Darboux transformation for its spectral problems is found. As an application, explicit solutions of the （2＋1）-dimensional Toda equation with two discrete variables are obtained.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11705077 and 11775104Natural Science Foundation of Zhejiang Province under Grant No.LY14A010005Scientific Research Foundation of the First-Class Discipline of Zhejiang Province(B)(No.201601)
文摘Nonlocal symmetry and explicit solution of the integrable Alice-Bob modified Korteweg-de Vries(ABm Kd V) equation is discussed, which has been established by the aid of the shifted parity and delayed time reversal to describe two-place events. Based on the Lax pair which contains the two-order partial derivative, the Lie symmetry group method is successfully applied to find the exact invariant solution for the AB-m Kd V equation with nonlocal symmetry by introducing one suitable auxiliary variable. Meanwhile, based on the prolonged system, the explicit analytic interaction solutions related to some specific functions are derived. Figures show the physical phenomenon, that is, "the shifted parity and delayed time reversal to describe two-place events".
文摘The generalized Zakharov equation is a coupled equation which is a classic nonlinear mathematic model in plasma. A series of new exact explicit solutions of the system are obtained, by means of the first integral method, in the form of trigonometric and exponential functions. The results show the first integral method is an efficient way to solve the coupled nonlinear equations and get rich explicit analytical solutions.
基金supported by the Mathematics and Physics Foundation of Beijing Polytechnic University and the National Natural Science Foundation of China (Grant No 40536029)
文摘Explicit solutions are derived for some nonlinear physical model equations by using a delicate way of two-step ansatz method.
