In this paper, new basic functions, which are composed of three basic Jacobi elliptic functions, are chosen as components of finite expansion. This finite expansion can be taken as an ansatz and applied to solve nonli...In this paper, new basic functions, which are composed of three basic Jacobi elliptic functions, are chosen as components of finite expansion. This finite expansion can be taken as an ansatz and applied to solve nonlinear wave equations. As an example, mKdV equation is solved, and more new rational form solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form, and so on.展开更多
A weakly nonholonomic system is a nonholonomic system whose constraint equations contain a small parameter. The form invariance and the approximate conserved quantity of the Appell equations for a weakly nonholonomic ...A weakly nonholonomic system is a nonholonomic system whose constraint equations contain a small parameter. The form invariance and the approximate conserved quantity of the Appell equations for a weakly nonholonomic system are studied. The Appell equations for the weakly nonholonomic system are established, and the definition and the criterion of form invariance of the system are given. The structural equation of form invariance for the weakly nonholonomic system and the approximate conserved quantity deduced from the form invariance of the system are obtained. Finally, an example is given to illustrate the application of the results.展开更多
We use Hopf-Lax formula to study local regularity of solution to Hamilton- Jacobi （HJ） equations of multi-dimensional space variables with convex Hamiltonian. Then we give the large time generic form of the solution...We use Hopf-Lax formula to study local regularity of solution to Hamilton- Jacobi （HJ） equations of multi-dimensional space variables with convex Hamiltonian. Then we give the large time generic form of the solution to HJ equation, i.e. for most initial data there exists a constant T 〉 0, which depends only on the Hamiltonian and initial datum, for t 〉 T the solution of the IVP （1.1） is smooth except for ~ smooth n-dimensional hypersurface, across which Du（x, t） is discontinuous. And we show that the hypersurface 1 tends asymptotically to a given hypersurface with rate t-1/4.展开更多
In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinea...In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems （with same stiffness matrix but different right-hand side） on the fine mesh space. The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the Mgorithm produces a numerical solution with the optimal asymptotic H^2-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.展开更多
The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical...The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical example of an integrable evolution equation in one spatial dimension.Do there exist integrable analogs of the modified Kd V equation in higher spatial dimensions?In what follows,we present a positive answer to this question.In particular,rewriting the(1+1)-dimensional integrable modified Kd V equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations.Further,we illustrate this idea with examples from the modified Kd V hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.展开更多
In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution ...In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution to the special case for z=x.Furthermore,a more general form of lump solution of the equation is found which possesses seven arbitrary parameters and four constraint conditions.By cutting the lump by the induced soliton(s),lumpoff and instanton/rogue wave solutions are also constructed by the more general form of lump solution.展开更多
In this paper, the investigation is focused on a （3＋1）-dimensional variable-coefficient Kadomtsev- Petviashvili （vcKP） equation, which can describe the realistic nonlinear phenomena in the fluid dynamics and plas...In this paper, the investigation is focused on a （3＋1）-dimensional variable-coefficient Kadomtsev- Petviashvili （vcKP） equation, which can describe the realistic nonlinear phenomena in the fluid dynamics and plasma in three spatial dimensions. In order to study the integrability property of such an equation, the Painlevé analysis is performed on it. And then, based on the truncated Painlevé expansion, the bilinear form of the （3＋1）-dimensionaJ vcKP equation is obtained under certain coefficients constraint, and its solution in the Wronskian determinant form is constructed and verified by virtue of the Wronskian technique. Besides the Wronskian determinant solution, it is shown that the （3＋1）-dimensional vcKP equation also possesses a solution in the form of the Grammian determinant.展开更多
In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations....In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.展开更多
In this paper, two types of the （2＋1）-dimensional breaking soliton equations axe investigated, which describe the interactions of the Riemann waves with the long waves. With symbolic computation, the Hirota bilinea...In this paper, two types of the （2＋1）-dimensional breaking soliton equations axe investigated, which describe the interactions of the Riemann waves with the long waves. With symbolic computation, the Hirota bilineax forms and Bgcklund transformations are derived for those two systems. Furthermore, multisoliton solutions in terms of the Wronskian determinant are constructed, which are verified through the direct substitution of the solutions into the bilineax equations. Via the Wronskian technique, it is proved that the Bgcklund transformations obtained are the ones between the （ N - 1）- and N-soliton solutions. Propagations and interactions of the kink-/bell-shaped solitons are presented. It is shown that the Riemann waves possess the solitonie properties, and maintain the amplitudes and velocities in the collisions only with some phase shifts.展开更多
The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized...The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized by a second-order finite difference procedure.The stability and the convergence of the semi-discrete formula have been proven.Then,the spatial variable of the main PDEs is approximated by the spectral element method.The convergence order of the fully discrete scheme is studied.The basis functions of the spectral element method are based upon a class of Legendre polynomials.The numerical experiments confirm the theoretical results.展开更多
In this paper, we show a general procedure to nonlinearize bilinear equations by using the Bell polynomials. As applications, we obtain nonlinear forms of some integrable bilinear equations(in the sense of having thre...In this paper, we show a general procedure to nonlinearize bilinear equations by using the Bell polynomials. As applications, we obtain nonlinear forms of some integrable bilinear equations(in the sense of having three-soliton solutions) of the KdV type and mKdV type that were found by Jarmo Hietarinta in the 1980s. Examples of non-integrable bilinear equations of the KdV type are also given.展开更多
In this paper the wood was regarded as the thermodynamie system. Using solid theory of crystalline state and thermodynamic theory, based on the expression of wood thermal capacity theory, the basic thermodynamic funct...In this paper the wood was regarded as the thermodynamie system. Using solid theory of crystalline state and thermodynamic theory, based on the expression of wood thermal capacity theory, the basic thermodynamic function of wood was derived. This paper emphasized the state equation of wood, brought it into Greuneisen form and comparcd the thcorctical value with the experimental value. And the relative error was in about 5%. This indicatcd that the results of the theoretical studies were right, explored a new way on the quantitative study of the physical characters of wood.展开更多
By using the Smith normal form of polynomial matrix and algebraic methods, this paper discusses the solvability for the linear matrix equation A iXB i=C over a field, and obtains the explicit formulas of general sol...By using the Smith normal form of polynomial matrix and algebraic methods, this paper discusses the solvability for the linear matrix equation A iXB i=C over a field, and obtains the explicit formulas of general solution or unique solution.展开更多
Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level met...Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.展开更多
In this paper,we derive the Lindblad and Redfield forms of the master equation based on the Born–Markov master equation with and without the secular approximation for open multi-level quantum systems.The coefficients...In this paper,we derive the Lindblad and Redfield forms of the master equation based on the Born–Markov master equation with and without the secular approximation for open multi-level quantum systems.The coefficients of the equations are re-evaluated according to the scheme in[(2019),Phys.Rev.A 99,022118].They are complex numbers rather than the real numbers obtained from traditional simplified methods.The dynamics of two models(one is an open threelevel quantum system model,and the other is the model of phycoerythrin 545(PE545)in a photosynthesis system)are studied.It is shown that the secular approximation and the simplified real coefficients may cause a small distortion of the dynamics in some environments,but a large distortion of the dynamics in others.These effects are discussed and characterized by studying the dynamics of nontrivial instances of multi-level systems in the presence of dissipation.展开更多
In this paper,the density-independent fractional diffusion-reaction(FDR)equation involving quadratic nonlinearity is investigated.The fractional derivative is illustrated in the beta derivative sense.We firstly propos...In this paper,the density-independent fractional diffusion-reaction(FDR)equation involving quadratic nonlinearity is investigated.The fractional derivative is illustrated in the beta derivative sense.We firstly propose Bernoulli(G'/G)-expansion method to study nonlinear fractional differential equations(NFDEs).Subsequently,closed form solutions of the density-independent FDR equation are acquired successfully.In order to better understand the dynamic behaviors of these solutions,3D,contour map and line plots are given by the computer simulation.The results show that the proposed method is a reliable and efficient approach.展开更多
The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown th...The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown that more novel kinds of solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form,and so on.展开更多
We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation l...We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation language M ATHEMATICA,and extending the matrix representation method.This method can be used to calculate the focal value of any high order terms.This method has been verified by an example.The advantage of this method is simple and more readily applicable.the result is directly obtained by substitution.展开更多
We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across th...We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners.展开更多
基金The project supported by National Natural Science Foundation of China under Grant No.40305006the Ministry of Science and Technology of China through Special Public Welfare Project under Grant No.2002DIB20070
文摘In this paper, new basic functions, which are composed of three basic Jacobi elliptic functions, are chosen as components of finite expansion. This finite expansion can be taken as an ansatz and applied to solve nonlinear wave equations. As an example, mKdV equation is solved, and more new rational form solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form, and so on.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11142014 and 61178032)
文摘A weakly nonholonomic system is a nonholonomic system whose constraint equations contain a small parameter. The form invariance and the approximate conserved quantity of the Appell equations for a weakly nonholonomic system are studied. The Appell equations for the weakly nonholonomic system are established, and the definition and the criterion of form invariance of the system are given. The structural equation of form invariance for the weakly nonholonomic system and the approximate conserved quantity deduced from the form invariance of the system are obtained. Finally, an example is given to illustrate the application of the results.
基金supported by National Natural Science Foundation of China (10871133,11071246 and 11101143)Fundamental Research Funds of the Central Universities (09QL48)
文摘We use Hopf-Lax formula to study local regularity of solution to Hamilton- Jacobi （HJ） equations of multi-dimensional space variables with convex Hamiltonian. Then we give the large time generic form of the solution to HJ equation, i.e. for most initial data there exists a constant T 〉 0, which depends only on the Hamiltonian and initial datum, for t 〉 T the solution of the IVP （1.1） is smooth except for ~ smooth n-dimensional hypersurface, across which Du（x, t） is discontinuous. And we show that the hypersurface 1 tends asymptotically to a given hypersurface with rate t-1/4.
基金supported by National Foundation of Natural Science under the Grant 11071216
文摘In this paper, we propose a two-grid algorithm for solving the stream function formulation of the stationary Navies-Stokes equations. The algorithm is constructed by reducing the original system to one small, nonlinear system on the coarse mesh space and two similar linear systems （with same stiffness matrix but different right-hand side） on the fine mesh space. The convergence analysis and error estimation of the algorithm are given for the case of conforming elements. Furthermore, the Mgorithm produces a numerical solution with the optimal asymptotic H^2-error. Finally, we give a numerical illustration to demonstrate the effectiveness of the two-grid algorithm for solving the Navier-Stokes equations.
基金sponsored by the National Natural Science Foundations of China(Nos.12235007,11975131,11435005,12275144,11975204)KC Wong Magna Fund in Ningbo UniversityNatural Science Foundation of Zhejiang Province No.LQ20A010009。
文摘The derivation of nonlinear integrable evolution partial differential equations in higher dimensions has always been the holy grail in the field of integrability.The well-known modified Kd V equation is a prototypical example of an integrable evolution equation in one spatial dimension.Do there exist integrable analogs of the modified Kd V equation in higher spatial dimensions?In what follows,we present a positive answer to this question.In particular,rewriting the(1+1)-dimensional integrable modified Kd V equation in conservation forms and adding deformation mappings during the process allows one to construct higher-dimensional integrable equations.Further,we illustrate this idea with examples from the modified Kd V hierarchy and also present the Lax pairs of these higher-dimensional integrable evolution equations.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11675084 and 11435005)the Fund from the Educational Commission of Zhejiang Province,China(Grant No.Y201737177)+1 种基金Ningbo Natural Science Foundation(Grant No.2015A610159)the K C Wong Magna Fund in Ningbo University
文摘In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution to the special case for z=x.Furthermore,a more general form of lump solution of the equation is found which possesses seven arbitrary parameters and four constraint conditions.By cutting the lump by the induced soliton(s),lumpoff and instanton/rogue wave solutions are also constructed by the more general form of lump solution.
基金Supported by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos. 20060006024 and 20080013006Chinese Ministry of Education, by the National Natural Science Foundation of China under Grant No. 60772023+2 种基金by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. SKLSDE-07-001Beijing University of Aeronautics and Astronauticsby the National Basic Research Program of China (973 Program) under Grant No. 2005CB321901
文摘In this paper, the investigation is focused on a （3＋1）-dimensional variable-coefficient Kadomtsev- Petviashvili （vcKP） equation, which can describe the realistic nonlinear phenomena in the fluid dynamics and plasma in three spatial dimensions. In order to study the integrability property of such an equation, the Painlevé analysis is performed on it. And then, based on the truncated Painlevé expansion, the bilinear form of the （3＋1）-dimensionaJ vcKP equation is obtained under certain coefficients constraint, and its solution in the Wronskian determinant form is constructed and verified by virtue of the Wronskian technique. Besides the Wronskian determinant solution, it is shown that the （3＋1）-dimensional vcKP equation also possesses a solution in the form of the Grammian determinant.
文摘In this work we devise an algebraic method to uniformly construct rational form solitary wave solutions and Jacobi and Weierstrass doubly periodic wave solutions of physical interest for nonlinear evolution equations. With the aid of symbolic computation, we apply the proposed method to solving the (1+1)-dimensional dispersive long wave equation and explicitly construct a series of exact solutions which include the rational form solitary wave solutions and elliptic doubly periodic wave solutions as special cases.
基金Supported by the National Natural Science Foundation of China under Grant No.60772023 the Open Fund under Grant No.BUAASKLSDE-09KF-04l+2 种基金Supported Project under Grant No.SKLSDE-2010ZX-07 of the State Key Laboratory of Software Development Environment,Beijing University of Aeronautics and Astronauticsthe National Basic Research Program of China (973 Program) under Grant No.2005CB321901 the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.200800130006,Chinese Ministry of Education
文摘In this paper, two types of the （2＋1）-dimensional breaking soliton equations axe investigated, which describe the interactions of the Riemann waves with the long waves. With symbolic computation, the Hirota bilineax forms and Bgcklund transformations are derived for those two systems. Furthermore, multisoliton solutions in terms of the Wronskian determinant are constructed, which are verified through the direct substitution of the solutions into the bilineax equations. Via the Wronskian technique, it is proved that the Bgcklund transformations obtained are the ones between the （ N - 1）- and N-soliton solutions. Propagations and interactions of the kink-/bell-shaped solitons are presented. It is shown that the Riemann waves possess the solitonie properties, and maintain the amplitudes and velocities in the collisions only with some phase shifts.
基金The authors are grateful to the two reviewers for carefully reading this paper and for their comments and suggestions which have highly improved the paper.
文摘The main aim of this paper is to analyze the numerical method based upon the spectral element technique for the numerical solution of the fractional advection-diffusion equa-tion.The time variable has been discretized by a second-order finite difference procedure.The stability and the convergence of the semi-discrete formula have been proven.Then,the spatial variable of the main PDEs is approximated by the spectral element method.The convergence order of the fully discrete scheme is studied.The basis functions of the spectral element method are based upon a class of Legendre polynomials.The numerical experiments confirm the theoretical results.
文摘In this paper, we show a general procedure to nonlinearize bilinear equations by using the Bell polynomials. As applications, we obtain nonlinear forms of some integrable bilinear equations(in the sense of having three-soliton solutions) of the KdV type and mKdV type that were found by Jarmo Hietarinta in the 1980s. Examples of non-integrable bilinear equations of the KdV type are also given.
文摘In this paper the wood was regarded as the thermodynamie system. Using solid theory of crystalline state and thermodynamic theory, based on the expression of wood thermal capacity theory, the basic thermodynamic function of wood was derived. This paper emphasized the state equation of wood, brought it into Greuneisen form and comparcd the thcorctical value with the experimental value. And the relative error was in about 5%. This indicatcd that the results of the theoretical studies were right, explored a new way on the quantitative study of the physical characters of wood.
基金the NSF of Hunan Province and the Science and Technology Development Foundation of Xiangtan Polytechnic University
文摘By using the Smith normal form of polynomial matrix and algebraic methods, this paper discusses the solvability for the linear matrix equation A iXB i=C over a field, and obtains the explicit formulas of general solution or unique solution.
文摘Residual based on a posteriori error estimates for conforming finite element solutions of incompressible Navier-Stokes equations with stream function form which were computed with seven recently proposed two-level method were derived. The posteriori error estimates contained additional terms in comparison to the error estimates for the solution obtained by the standard finite element method. The importance of these additional terms in the error estimates was investigated by studying their asymptotic behavior. For optimal scaled meshes, these bounds are not of higher order than of convergence of discrete solution.
文摘In this paper,we derive the Lindblad and Redfield forms of the master equation based on the Born–Markov master equation with and without the secular approximation for open multi-level quantum systems.The coefficients of the equations are re-evaluated according to the scheme in[(2019),Phys.Rev.A 99,022118].They are complex numbers rather than the real numbers obtained from traditional simplified methods.The dynamics of two models(one is an open threelevel quantum system model,and the other is the model of phycoerythrin 545(PE545)in a photosynthesis system)are studied.It is shown that the secular approximation and the simplified real coefficients may cause a small distortion of the dynamics in some environments,but a large distortion of the dynamics in others.These effects are discussed and characterized by studying the dynamics of nontrivial instances of multi-level systems in the presence of dissipation.
基金Supported by the National Natural Science Foundation of China(11901111)Guangzhou Science and Technology Plan Project(202201011602)。
文摘In this paper,the density-independent fractional diffusion-reaction(FDR)equation involving quadratic nonlinearity is investigated.The fractional derivative is illustrated in the beta derivative sense.We firstly propose Bernoulli(G'/G)-expansion method to study nonlinear fractional differential equations(NFDEs).Subsequently,closed form solutions of the density-independent FDR equation are acquired successfully.In order to better understand the dynamic behaviors of these solutions,3D,contour map and line plots are given by the computer simulation.The results show that the proposed method is a reliable and efficient approach.
文摘The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown that more novel kinds of solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form,and so on.
文摘We present a new method to calculate the focal value of ordinary differential equation by applying the theorem defined the relationship between the normal form and focal value,with the help of a symbolic computation language M ATHEMATICA,and extending the matrix representation method.This method can be used to calculate the focal value of any high order terms.This method has been verified by an example.The advantage of this method is simple and more readily applicable.the result is directly obtained by substitution.
基金supported by National Natural Science Foundation of China(12061080,12161087 and 12261093)the Science and Technology Project of the Education Department of Jiangxi Province(GJJ211601)supported by National Natural Science Foundation of China(11871305).
文摘We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners.
