The existence and uniqueness of classical global solutions and the nonexistence of global solutions to the first boundary value problem and the second boundary value problem for the equation u tt -a 1u xx -a ...The existence and uniqueness of classical global solutions and the nonexistence of global solutions to the first boundary value problem and the second boundary value problem for the equation u tt -a 1u xx -a 2u xxt -a 3u xxtt =φ(u x ) x are proved.展开更多
Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are pro...Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are proved by the energy method, Jensen inequality and the concavity method, respectively. As applications of our main results, three examples are given.展开更多
In this paper, the existence and uniqueness of the local generalized solution of the initial boundary value problem for a nonlinear hyperbolic equation are proved by the contraction mapping principle and the sufficien...In this paper, the existence and uniqueness of the local generalized solution of the initial boundary value problem for a nonlinear hyperbolic equation are proved by the contraction mapping principle and the sufficient conditions of blow_up of the solution in finite time are given.展开更多
A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the pa...A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the parameter can be applied in judging the existence of various forms of travelling wave solutions.An efficiency of this method is demonstrated on some equations,which include Burgers Huxley equation,Caudrey Dodd Gibbon Kawada equation,generalized Benjamin Bona Mahony equation and generalized Fisher equation.展开更多
A unified approach is presented for finding the travelling wave solutions to one kind of nonlinear evolution equation by introducing a concept of 'rank'. The key idea of this method is to make use of the arbit...A unified approach is presented for finding the travelling wave solutions to one kind of nonlinear evolution equation by introducing a concept of 'rank'. The key idea of this method is to make use of the arbitrariness of the manifold in Painlevé analysis. We selected a new expansion variable and thus obtained a rich variety of travelling wave solutions to nonlinear evolution equation, which covered solitary wave solutions, periodic wave solutions, Weierstrass elliptic function solutions, and rational solutions. Three illustrative equations are investigated by this means, and abundant travelling wave solutions are obtained in a systematic way. In addition, some new solutions are firstly reported here.展开更多
To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are pr...To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.展开更多
We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional sep...We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting 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
Instead of the usual Hirota ansatz,i.e.,the functions in bilinear equations being chosen as exponentialtypes,a generalized Hirota ansatz is proposed for a (3+1)-dimensional nonlinear evolution equation.Based on theres...Instead of the usual Hirota ansatz,i.e.,the functions in bilinear equations being chosen as exponentialtypes,a generalized Hirota ansatz is proposed for a (3+1)-dimensional nonlinear evolution equation.Based on theresulting generalized Hirota ansatz,a family of new explicit solutions for the equation are derived.展开更多
Using Daher's fixed point theorem, we obtain a local existence theorem, in which the assumption is weaker than That in the Theorem 2.1 in [2]. Based on this theorem, we get a global existence theorem which is an e...Using Daher's fixed point theorem, we obtain a local existence theorem, in which the assumption is weaker than That in the Theorem 2.1 in [2]. Based on this theorem, we get a global existence theorem which is an extension of certain results for ordinary differential equations.展开更多
In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new typ...In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new types of periodic wave solutions which are rarely found in previous studies. As <em>m</em> → 0 and <em>m</em> → 1, some new types of trigonometric solutions and solitary solutions are also obtained correspondingly. This method is promising for constructing abundant periodic wave solutions and solitary solutions of nonlinear evolution equations (NLEEs) in mathematical physics.展开更多
The homogeneous balance method was improved and applied to two systems Of nonlinear evolution equations. As a result, several families of exact analytic solutions are derived by some new ansatzs. These solutions conta...The homogeneous balance method was improved and applied to two systems Of nonlinear evolution equations. As a result, several families of exact analytic solutions are derived by some new ansatzs. These solutions contain Wang's and Zhang's results and other new types of analytical solutions, such as rational fraction solutions and periodic solutions. The way can also be applied to solve more nonlinear partial differential equations.展开更多
By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave...By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave solutions and uncountably infinite many smooth solitary wave solutions are given.展开更多
We applied the multiple exp-function scheme to the（2＋1）-dimensional Sawada-Kotera（SK） equation and（3＋1）-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analyti...We applied the multiple exp-function scheme to the（2＋1）-dimensional Sawada-Kotera（SK） equation and（3＋1）-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota＇s perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.展开更多
This paper gives the suffcient conditions of blow-up of the solution of a nonlinear hyperbolic equation with the initial boundary value conditions in finite time and proves the existence and uniqueness of the local so...This paper gives the suffcient conditions of blow-up of the solution of a nonlinear hyperbolic equation with the initial boundary value conditions in finite time and proves the existence and uniqueness of the local solution of the problem.展开更多
In this paper, the Adomian decomposition method is developed for the numerical solutions of a class of nonlinear evolution equations with nonlinear term of any order, utt＋auxx ＋ bu ＋ cu^p＋ du^2p-1=0, which contain...In this paper, the Adomian decomposition method is developed for the numerical solutions of a class of nonlinear evolution equations with nonlinear term of any order, utt＋auxx ＋ bu ＋ cu^p＋ du^2p-1=0, which contains some important famous equations. When setting the initial conditions in different forms, some new generalized numerical solutions： numerical hyperbolic solutions, numerical doubly periodic solutions are obtained. The numerical solutions are compared with exact solutions. The scheme is tested by choosing different values of p, positive and negative, integer and fraction, to illustrate the efficiency of the ADM method and the generalization of the solutions.展开更多
This paper is based on the relations between projection Riccati equations and Weierstrass elliptic equation, combined with the Groebner bases in the symbolic computation.Then the novel method for constructing the Weie...This paper is based on the relations between projection Riccati equations and Weierstrass elliptic equation, combined with the Groebner bases in the symbolic computation.Then the novel method for constructing the Weierstrass elliptic solutions to the nonlinear evolution equations is given by using the above relations.展开更多
Based on the Hirota bilinear form, a simple approach without employing the standard perturbation technique, is presented for constructing a novel N-soliton solution for a （3＋1）-dimensional nonlinear evolution equat...Based on the Hirota bilinear form, a simple approach without employing the standard perturbation technique, is presented for constructing a novel N-soliton solution for a （3＋1）-dimensional nonlinear evolution equation. Moreover, the novel N-soliton solution is shown to have resonant behavior with the aid of Mathematica.展开更多
J. L Lions and W. A. Stranss [1] have proved the existence of a global solution of the initial boundary value problem for nonlinear generalized Euler-Possion-Darboux equation. In this paper we are going to investigate...J. L Lions and W. A. Stranss [1] have proved the existence of a global solution of the initial boundary value problem for nonlinear generalized Euler-Possion-Darboux equation. In this paper we are going to investigate the asymptotic behavior of the global solution by a difference inequality.展开更多
基金the National Natural Science Foundation of China(1 0 0 71 0 74) and the Natural ScienceFoundation of Henan Provinc
文摘The existence and uniqueness of classical global solutions and the nonexistence of global solutions to the first boundary value problem and the second boundary value problem for the equation u tt -a 1u xx -a 2u xxt -a 3u xxtt =φ(u x ) x are proved.
基金Project supported by the National Natural Science Foundation of China (Nos. 10371073 and 10572156) the Natural Science Foundation of Henan Province of China (No.0611050500)
文摘Concerns with the nonexistence of global solutions to the initial boundary value problem for a nonlinear hyperbolic equation with material damping. Nonexitence theorems of global solutions to the above problem are proved by the energy method, Jensen inequality and the concavity method, respectively. As applications of our main results, three examples are given.
文摘In this paper, the existence and uniqueness of the local generalized solution of the initial boundary value problem for a nonlinear hyperbolic equation are proved by the contraction mapping principle and the sufficient conditions of blow_up of the solution in finite time are given.
基金Supported by the Postdoctoral Science Foundation of ChinaChinese Basic Research Plan"MathematicsMechanization and A Platform
文摘A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the parameter can be applied in judging the existence of various forms of travelling wave solutions.An efficiency of this method is demonstrated on some equations,which include Burgers Huxley equation,Caudrey Dodd Gibbon Kawada equation,generalized Benjamin Bona Mahony equation and generalized Fisher equation.
文摘A unified approach is presented for finding the travelling wave solutions to one kind of nonlinear evolution equation by introducing a concept of 'rank'. The key idea of this method is to make use of the arbitrariness of the manifold in Painlevé analysis. We selected a new expansion variable and thus obtained a rich variety of travelling wave solutions to nonlinear evolution equation, which covered solitary wave solutions, periodic wave solutions, Weierstrass elliptic function solutions, and rational solutions. Three illustrative equations are investigated by this means, and abundant travelling wave solutions are obtained in a systematic way. In addition, some new solutions are firstly reported here.
基金supported by the National Natural Science Foundation of China(Grant No.10862003)the Science Research Foundation of Institution of Higher Education of Inner Mongolia Autonomous Region,China(Grant No.NJZZ07031)the Natural Science Foundation of Inner Mongolia Autonomous Region,China(Grant No.2010MS0111)
文摘To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.
基金The project supported by 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We give the generalized definitions of variable separable solutions to nonlinear evolution equations, and characterize the relation between the functional separable solution and the derivative-dependent functional separable solution. The new definitions can unify various kinds of variable separable solutions appearing in references. As application, we classify the generalized nonlinear diffusion equations that admit special functional separable solutions and obtain some exact solutions to the resulting 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.
文摘Instead of the usual Hirota ansatz,i.e.,the functions in bilinear equations being chosen as exponentialtypes,a generalized Hirota ansatz is proposed for a (3+1)-dimensional nonlinear evolution equation.Based on theresulting generalized Hirota ansatz,a family of new explicit solutions for the equation are derived.
文摘Using Daher's fixed point theorem, we obtain a local existence theorem, in which the assumption is weaker than That in the Theorem 2.1 in [2]. Based on this theorem, we get a global existence theorem which is an extension of certain results for ordinary differential equations.
文摘In this paper, a new auxiliary equation method is proposed. Combined with the mapping method, abundant periodic wave solutions for generalized Klein-Gordon equation and Benjamin equation are obtained. They are new types of periodic wave solutions which are rarely found in previous studies. As <em>m</em> → 0 and <em>m</em> → 1, some new types of trigonometric solutions and solitary solutions are also obtained correspondingly. This method is promising for constructing abundant periodic wave solutions and solitary solutions of nonlinear evolution equations (NLEEs) in mathematical physics.
文摘The homogeneous balance method was improved and applied to two systems Of nonlinear evolution equations. As a result, several families of exact analytic solutions are derived by some new ansatzs. These solutions contain Wang's and Zhang's results and other new types of analytical solutions, such as rational fraction solutions and periodic solutions. The way can also be applied to solve more nonlinear partial differential equations.
基金Project supported by the National Natural Science Foundation of China(Nos.10671179,10772158)
文摘By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave solutions and uncountably infinite many smooth solitary wave solutions are given.
文摘We applied the multiple exp-function scheme to the（2＋1）-dimensional Sawada-Kotera（SK） equation and（3＋1）-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota＇s perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.
基金Supported by the National Natural Science Foundation of China（10671182） Supported by the Excellent Youth Teachers Foundation of High College of Henan Province（2006110016）
文摘This paper gives the suffcient conditions of blow-up of the solution of a nonlinear hyperbolic equation with the initial boundary value conditions in finite time and proves the existence and uniqueness of the local solution of the problem.
基金supported by National Natural Science Foundation of China under Grant No.10735030Natural Science Foundation of Zhejiang Province of China under Grant No.Y604056Doctoral Science Foundation of Ningbo City under Grant No.2005A61030
文摘In this paper, the Adomian decomposition method is developed for the numerical solutions of a class of nonlinear evolution equations with nonlinear term of any order, utt＋auxx ＋ bu ＋ cu^p＋ du^2p-1=0, which contains some important famous equations. When setting the initial conditions in different forms, some new generalized numerical solutions： numerical hyperbolic solutions, numerical doubly periodic solutions are obtained. The numerical solutions are compared with exact solutions. The scheme is tested by choosing different values of p, positive and negative, integer and fraction, to illustrate the efficiency of the ADM method and the generalization of the solutions.
基金supported by the Open Project of Key Laboratory of Mathematics Mechanization,CAS under Grant No.KLMM0602
文摘This paper is based on the relations between projection Riccati equations and Weierstrass elliptic equation, combined with the Groebner bases in the symbolic computation.Then the novel method for constructing the Weierstrass elliptic solutions to the nonlinear evolution equations is given by using the above relations.
文摘Based on the Hirota bilinear form, a simple approach without employing the standard perturbation technique, is presented for constructing a novel N-soliton solution for a （3＋1）-dimensional nonlinear evolution equation. Moreover, the novel N-soliton solution is shown to have resonant behavior with the aid of Mathematica.
文摘J. L Lions and W. A. Stranss [1] have proved the existence of a global solution of the initial boundary value problem for nonlinear generalized Euler-Possion-Darboux equation. In this paper we are going to investigate the asymptotic behavior of the global solution by a difference inequality.
