An algebraic method with symbolic computation is devised to uniformly construct a series of exact solutions of the （2＋1）-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawda equation. The solutions obtained in this paper i...An algebraic method with symbolic computation is devised to uniformly construct a series of exact solutions of the （2＋1）-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawda equation. The solutions obtained in this paper include solitary wave solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic solutions. Among them, the Jacobi periodic solutions exactly degenerate to the solutions at a certain limit condition. Compared with most existing tanh method, the method used here can give new and more general solutions. More importantly, this method provides a guldeline to classifj, the various types of the solution according to some parameters.展开更多
We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. By using extended F-expansion method, many periodic wave solutions expressed by v...We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. By using extended F-expansion method, many periodic wave solutions expressed by various Jacobi elliptic functions for the Klein-Gordon-Schrodinger equations are obtained. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.展开更多
Abstract In this paper,the existence of periodic solution for the third order nonlinear ordinary differential equation of the form x…+f()+g()+h(x)=p(t) is considered,where f,g,h and p are the continuous f...Abstract In this paper,the existence of periodic solution for the third order nonlinear ordinary differential equation of the form x…+f()+g()+h(x)=p(t) is considered,where f,g,h and p are the continuous functions,and p(t+T)=p(t). By using the Leray Schauder degree method,some sufficient conditions to guarantee the existence of T periodic solution for the equation are obtained.Some previous results are extended and improved.展开更多
In this paper, a series of two line-soliton solutions and double periodic solutions of Chaffee-Infante equation have been obtained by using a new transformation. Unlike the existing methods which are used to find mult...In this paper, a series of two line-soliton solutions and double periodic solutions of Chaffee-Infante equation have been obtained by using a new transformation. Unlike the existing methods which are used to find multiple soliton solutions of nonlinear partial differential equations, this approach is constructive and pure algebraic. The results found here are tested on computer and therefore their validity is ensured.展开更多
Trial function method is applied to solve generalized mKdV (GmKdV for short) equations. It is shownthat GmKdV equations with a real number parameter can be solved directly by this method without a transformation,and m...Trial function method is applied to solve generalized mKdV (GmKdV for short) equations. It is shownthat GmKdV equations with a real number parameter can be solved directly by this method without a transformation,and more new kinds of solitary wave solutions are obtained.展开更多
In this paper, many new explicit and exact travelling wave solutions for Burgers-Kolmogorov-Petrovskii-Piscounov(Burgers-KPP) equations are obtained by using hyperbola function method and Wu-elimination method, which ...In this paper, many new explicit and exact travelling wave solutions for Burgers-Kolmogorov-Petrovskii-Piscounov(Burgers-KPP) equations are obtained by using hyperbola function method and Wu-elimination method, which include new singular solitary wave solutions and periodic solutions. Particular important cases of the equation, such as the generalized Burgers-Fisher equation, Burgers-Chaffee infante equation and KPP equation, the corresponding solutions can be obtained also. The method can also solve other nonlinear partial differential equations.展开更多
In this work, an adaptation of the tanh/tan-method that is discussed usually in the nonlinear partial differential equations is presented to solve nonlinear polynomial differential-difference equations. As a concrete ...In this work, an adaptation of the tanh/tan-method that is discussed usually in the nonlinear partial differential equations is presented to solve nonlinear polynomial differential-difference equations. As a concrete example,several solitary wave and periodic wave solutions for the chain which is related to the relativistic Toda lattice are derived.Some systems of the differential-difference equations that can be solved using our approach are listed and a discussion is given in conclusion.展开更多
Based on computerized symbolic computation,a new method and its algorithm are proposed for searching for exact travelling wave solutions of the nonlinear partial differential equations.Making use of our approach,we in...Based on computerized symbolic computation,a new method and its algorithm are proposed for searching for exact travelling wave solutions of the nonlinear partial differential equations.Making use of our approach,we investigate the Whitham-Broer-Kaup equation in shallow water and obtain new families of exact solutions,which include soliton-like solutions and periodic solutions.As its special cases,the solutions of classical long wave equations and modified Boussinesq equations can also be found.展开更多
A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, th...A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.展开更多
This paper applies the EXP-function method to find exact solutions of various nonlinear equations. Tzitzeica- Dodd-Bullough (TDB) equation was selected to illustrate the effectiveness and convenience of the suggested ...This paper applies the EXP-function method to find exact solutions of various nonlinear equations. Tzitzeica- Dodd-Bullough (TDB) equation was selected to illustrate the effectiveness and convenience of the suggested method. More generalized solitonary solutions with free parameters were obtained by suitable choice of the free parameters, and also the obtained solitonary solutions can be converted into periodic solutions.展开更多
By using the general solutions of a new coupled Riccati equations, a direct algebraic method is described to construct doubly periodic solutions (Jacobi elliptic function solution) for the coupled nonlinear Klein-Gord...By using the general solutions of a new coupled Riccati equations, a direct algebraic method is described to construct doubly periodic solutions (Jacobi elliptic function solution) for the coupled nonlinear Klein-Gordon equations.It is shown that more doubly periodic solutions and the corresponding solitary wave solutions and trigonometric function solutions can be obtained in a unified way by this method.展开更多
In this paper, the generalized ranch function method is extended to （2＋1）-dimensianal canonical generalized KP （CGKP） equation with variable coetfficients. Taking advantage of the Riccati equation, many explicit ...In this paper, the generalized ranch function method is extended to （2＋1）-dimensianal canonical generalized KP （CGKP） equation with variable coetfficients. Taking advantage of the Riccati equation, many explicit exact solutions, which contain multiple soliton-like and periodic solutions, are obtained for the （2＋1）-dimensional OGKP equation with variable coetffcients.展开更多
A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic w...A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions, and other exact excitations like polynomial solutions, exponential solutions, and rational solutions, etc., are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and a generalized cubic nonlinear Klein-Gordon equation.展开更多
With the aid of computerized symbolic computation, the new modified Jacobi elliptic function expansion method for constructing exact periodic solutions of nonlinear mathematical physics equation is presented by a new ...With the aid of computerized symbolic computation, the new modified Jacobi elliptic function expansion method for constructing exact periodic solutions of nonlinear mathematical physics equation is presented by a new general ansatz. The proposed method is more powerful than most of the existing methods. By use of the method, we not only can successfully recover the previously known formal solutions but also can construct new and more general formal solutions for some nonlinear evolution equations. We choose the （3＋1）-dimensional Kadomtsev-Petviashvili equation to illustrate our method. As a result, twenty families of periodic solutions are obtained. Of course, more solitary wave solutions, shock wave solutions or triangular function formal solutions can be obtained at their limit condition.展开更多
By using exponential dichotomies and Liapunov function method, we have studied the existence of almost periodic solutions on a Lienard system and have obtained some simple sufficient condition.
By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified wa...By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m → 1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.展开更多
For describing various complex nonlinear phenomena in the realistic world,the higher-dimensional nonlinearevolution equations appear more attractive in many fields of physical and engineering sciences.In this paper,by...For describing various complex nonlinear phenomena in the realistic world,the higher-dimensional nonlinearevolution equations appear more attractive in many fields of physical and engineering sciences.In this paper,by virtueof the Hirota bilinear method and Riemann theta functions,the periodic wave solutions for the(2+1)-dimensionalBoussinesq equation and(3+1)-dimensional Kadomtsev-Petviashvili(KP)equation are obtained.Furthermore,it isshown that the known soliton solutions for the two equations can be reduced from the periodic wave solutions.展开更多
In this paper, applying the dependent and independent variables transformations as well as the Jacobi elliptic function expansion method, the envelope periodic solutions to one-dimensional Gross-Pitaevskii equation in...In this paper, applying the dependent and independent variables transformations as well as the Jacobi elliptic function expansion method, the envelope periodic solutions to one-dimensional Gross-Pitaevskii equation in Bose-Einstein condensates are obtained.展开更多
We present an approximate analytical solution to periodically driven two-level system in the weak-coupling regime. The analytical solution is in good agreement with the exact numerical solution in resonance and near r...We present an approximate analytical solution to periodically driven two-level system in the weak-coupling regime. The analytical solution is in good agreement with the exact numerical solution in resonance and near resonance cases when Ω 〈 0.3ωa with Ω and ωa denoting the Rabi and transition frequencies respectively.展开更多
基金The project supported by the City University of Hong Kong Strategic Research under Grant No. 7001791
文摘An algebraic method with symbolic computation is devised to uniformly construct a series of exact solutions of the （2＋1）-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawda equation. The solutions obtained in this paper include solitary wave solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic solutions. Among them, the Jacobi periodic solutions exactly degenerate to the solutions at a certain limit condition. Compared with most existing tanh method, the method used here can give new and more general solutions. More importantly, this method provides a guldeline to classifj, the various types of the solution according to some parameters.
基金The project supported by the Natural Science Foundation of Eduction Committce of Henan Province of China under Grant No. 2003110003, and the Science Foundation of Henan University of Science and Technology under Grant Nos. 2004ZD002 and 2004ZY040
文摘We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. By using extended F-expansion method, many periodic wave solutions expressed by various Jacobi elliptic functions for the Klein-Gordon-Schrodinger equations are obtained. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.
文摘Abstract In this paper,the existence of periodic solution for the third order nonlinear ordinary differential equation of the form x…+f()+g()+h(x)=p(t) is considered,where f,g,h and p are the continuous functions,and p(t+T)=p(t). By using the Leray Schauder degree method,some sufficient conditions to guarantee the existence of T periodic solution for the equation are obtained.Some previous results are extended and improved.
基金The project supported by '973' Project under Grant No.2004CB318000Doctor Start-up Foundation of Liaoning Province under Grant No.1040225Science and Technology Research Project of Liaoning Education Bureau
文摘In this paper, a series of two line-soliton solutions and double periodic solutions of Chaffee-Infante equation have been obtained by using a new transformation. Unlike the existing methods which are used to find multiple soliton solutions of nonlinear partial differential equations, this approach is constructive and pure algebraic. The results found here are tested on computer and therefore their validity is ensured.
基金The project supported by National Natural Science Foundation of China under Grant Nos.40045016 and 40175016
文摘Trial function method is applied to solve generalized mKdV (GmKdV for short) equations. It is shownthat GmKdV equations with a real number parameter can be solved directly by this method without a transformation,and more new kinds of solitary wave solutions are obtained.
基金Supported by the National Key Basic Research Development Project of China(1998030600)Supported by the National Natural Science Foudation of China(10072013)Supported by the Educational Commmittee of Liaoning Province(990421093)
文摘In this paper, many new explicit and exact travelling wave solutions for Burgers-Kolmogorov-Petrovskii-Piscounov(Burgers-KPP) equations are obtained by using hyperbola function method and Wu-elimination method, which include new singular solitary wave solutions and periodic solutions. Particular important cases of the equation, such as the generalized Burgers-Fisher equation, Burgers-Chaffee infante equation and KPP equation, the corresponding solutions can be obtained also. The method can also solve other nonlinear partial differential equations.
文摘In this work, an adaptation of the tanh/tan-method that is discussed usually in the nonlinear partial differential equations is presented to solve nonlinear polynomial differential-difference equations. As a concrete example,several solitary wave and periodic wave solutions for the chain which is related to the relativistic Toda lattice are derived.Some systems of the differential-difference equations that can be solved using our approach are listed and a discussion is given in conclusion.
文摘Based on computerized symbolic computation,a new method and its algorithm are proposed for searching for exact travelling wave solutions of the nonlinear partial differential equations.Making use of our approach,we investigate the Whitham-Broer-Kaup equation in shallow water and obtain new families of exact solutions,which include soliton-like solutions and periodic solutions.As its special cases,the solutions of classical long wave equations and modified Boussinesq equations can also be found.
文摘A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.
文摘This paper applies the EXP-function method to find exact solutions of various nonlinear equations. Tzitzeica- Dodd-Bullough (TDB) equation was selected to illustrate the effectiveness and convenience of the suggested method. More generalized solitonary solutions with free parameters were obtained by suitable choice of the free parameters, and also the obtained solitonary solutions can be converted into periodic solutions.
文摘By using the general solutions of a new coupled Riccati equations, a direct algebraic method is described to construct doubly periodic solutions (Jacobi elliptic function solution) for the coupled nonlinear Klein-Gordon equations.It is shown that more doubly periodic solutions and the corresponding solitary wave solutions and trigonometric function solutions can be obtained in a unified way by this method.
基金The project supported by the Natural Science Foundation of Shandong Province under Grant Nos. 2004zx16 and Q2005A01
文摘In this paper, the generalized ranch function method is extended to （2＋1）-dimensianal canonical generalized KP （CGKP） equation with variable coetfficients. Taking advantage of the Riccati equation, many explicit exact solutions, which contain multiple soliton-like and periodic solutions, are obtained for the （2＋1）-dimensional OGKP equation with variable coetffcients.
文摘A general mapping deformation method is presented and applied to a (2+1)-dimensional Boussinesq system. Many new types of explicit and exact travelling wave solutions, which contain solitary wave solutions, periodic wave solutions, Jacobian and Weierstrass doubly periodic wave solutions, and other exact excitations like polynomial solutions, exponential solutions, and rational solutions, etc., are obtained by a simple algebraic transformation relation between the (2+1)-dimensional Boussinesq equation and a generalized cubic nonlinear Klein-Gordon equation.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 10247008 and 10575082, the Natural Science Foundation of Gansu Province of China under Grant No. YS021-A22-018, the Scientific Research Foundation for thc Returned 0verseas Chinese Scholars of the Ministry of Education
文摘With the aid of computerized symbolic computation, the new modified Jacobi elliptic function expansion method for constructing exact periodic solutions of nonlinear mathematical physics equation is presented by a new general ansatz. The proposed method is more powerful than most of the existing methods. By use of the method, we not only can successfully recover the previously known formal solutions but also can construct new and more general formal solutions for some nonlinear evolution equations. We choose the （3＋1）-dimensional Kadomtsev-Petviashvili equation to illustrate our method. As a result, twenty families of periodic solutions are obtained. Of course, more solitary wave solutions, shock wave solutions or triangular function formal solutions can be obtained at their limit condition.
文摘By using exponential dichotomies and Liapunov function method, we have studied the existence of almost periodic solutions on a Lienard system and have obtained some simple sufficient condition.
文摘By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m → 1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.
基金supported by the National Natural Science Foundation of China under Grant Nos.60772023 and 60372095the Key Project of the Ministry of Education under Grant No.106033+3 种基金the Open Fund of the State Key Laboratory of Software Development Environment under Grant No.SKLSDE-07-001Beijing University of Aeronautics and Astronauticsthe National Basic Research Program of China(973 Program)under Grant No.2005CB321901the Specialized Research Fund for the Doctoral Program of Higher Education of the Ministry of Education under Grant No.20060006024
文摘For describing various complex nonlinear phenomena in the realistic world,the higher-dimensional nonlinearevolution equations appear more attractive in many fields of physical and engineering sciences.In this paper,by virtueof the Hirota bilinear method and Riemann theta functions,the periodic wave solutions for the(2+1)-dimensionalBoussinesq equation and(3+1)-dimensional Kadomtsev-Petviashvili(KP)equation are obtained.Furthermore,it isshown that the known soliton solutions for the two equations can be reduced from the periodic wave solutions.
基金Supported by National Natural Science Foundation of China under Grant No. 90511009
文摘In this paper, applying the dependent and independent variables transformations as well as the Jacobi elliptic function expansion method, the envelope periodic solutions to one-dimensional Gross-Pitaevskii equation in Bose-Einstein condensates are obtained.
基金The project supported by the Postdoctoral Science Foundation of China under Grant Nos.20060408878 and 2007RS4015Key Science Research Foundation of the Education Ministry of China,Natural Science Foundation of Hunan Province of China under Grant No.05JJ40007Key Science Research Foundation of the Education Department of Hunan Province under Grant No.07A057
文摘We present an approximate analytical solution to periodically driven two-level system in the weak-coupling regime. The analytical solution is in good agreement with the exact numerical solution in resonance and near resonance cases when Ω 〈 0.3ωa with Ω and ωa denoting the Rabi and transition frequencies respectively.
