This paper concerns with the Cauchy problem for the nonlinear double dispersive wave equation. By the priori estimates and the method in [9], It proves that the Cauchy problem admits a unique global classical solution...This paper concerns with the Cauchy problem for the nonlinear double dispersive wave equation. By the priori estimates and the method in [9], It proves that the Cauchy problem admits a unique global classical solution. And by the concave method, we give sufficient conditions on the blowup of the global solution for the Cauchy problem.展开更多
In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichart...In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichartz's inequality with the commutator argument techniques, we show that the weak solutions stay globally conormal if the Cauchy data are conormal展开更多
The inverse problem for harmonic waves and wave packets was studied based on a full dispersive wave equation. First, a full dispersive wave equation which describes wave propagation in nondissipative microstructured l...The inverse problem for harmonic waves and wave packets was studied based on a full dispersive wave equation. First, a full dispersive wave equation which describes wave propagation in nondissipative microstructured linear solids is established based on the Mindlin theory, and the dispersion characteristics are discussed. Second, based on the full dispersive wave equation, an inverse problem for determining the four unknown coefficients of wave equa- tion is posed in terms of the frequencies and corresponding wave numbers of four different harmonic waves, and the inverse problem is demonstrated with rigorous mathematical theory. Research proves that the coefficients of wave equation related to material properties can be uniquely determined in cases of normal and anomalous dispersions by measuring the frequen- cies and corresponding wave numbers of four different harmonic waves which propagate in a nondissipative microstructured linear solids.展开更多
A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a (1+1)-dimensional dispersive long wave equation with general coefficients to an elementary integral fo...A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a (1+1)-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.展开更多
We investigate (2+1)-dimensional generalized modified dispersive water wave (GMDWW) equation by utilizing the bifurcation theory of dynamical systems. We give the phase portraits and bifurcation analysis of the plane ...We investigate (2+1)-dimensional generalized modified dispersive water wave (GMDWW) equation by utilizing the bifurcation theory of dynamical systems. We give the phase portraits and bifurcation analysis of the plane system corresponding to the GMDWW equation. By using the special orbits in the phase portraits, we analyze the existence of the traveling wave solutions. When some parameter takes special values, we obtain abundant exact kink wave solutions, singular wave solutions, periodic wave solutions, periodic singular wave solutions, and solitary wave solutions for the GMDWW equation.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
By using the homogeneous balance principle(HBP), we derive a Backtund transformation(BT) to the generalized dispersive long wave equation with variable coefficients.Based on the BT, we give many kinds of the exact...By using the homogeneous balance principle(HBP), we derive a Backtund transformation(BT) to the generalized dispersive long wave equation with variable coefficients.Based on the BT, we give many kinds of the exact solutions of the equation, such as, singlesolitary solutions, multi-soliton solutions and generalized exact solutions.展开更多
By means of two different Riccati equations with different parameters as subequation in the components of finite rational expansion method, new complexiton solutions for the (1+1)-dimensional dispersive long wave e...By means of two different Riccati equations with different parameters as subequation in the components of finite rational expansion method, new complexiton solutions for the (1+1)-dimensional dispersive long wave equation are successfully constructed, which include various combination of trigonometric periodic and hyperbolic function solutions, various combination of trigonometric periodic and rational function solutions, and various combination of hyperbolic and rational function solutions.展开更多
A new generalized extended F-expansion method is presented for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. As an application of this method, we study the (2+1)-dimensio...A new generalized extended F-expansion method is presented for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. As an application of this method, we study the (2+1)-dimensional dispersive long wave equation. With the aid of computerized symbolic computation, a number of doubly periodic wave solutions expressed by various Jacobi elliptic functions are obtained. In the limit cases, the solitary wave solutions are derived as well.展开更多
In this article, we study the Lax pairs of -dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion formula...In this article, we study the Lax pairs of -dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion formulas of the first part of the Lax pairs. Then by further discussion and doing some revisional work, we make the recursion formulas fit for the second part of Lax pairs. At last, some solutions to the MGDLW equation are worked out by using the recursion formula.展开更多
Seeking exact analytical solutions of nonlinear evolution equations is of fundamental importance in mathematlcal physics. In this paper, based on a constructive algorithm and symbolic computation, abundant new exact s...Seeking exact analytical solutions of nonlinear evolution equations is of fundamental importance in mathematlcal physics. In this paper, based on a constructive algorithm and symbolic computation, abundant new exact solutions of the (2+1)-dimensional dispersive long wave equations are obtained, among which there are soliton-like solutions, mult-soliton-like solutions and formal periodic solutions, etc. Certain special solutions are considered and some interesting localized structures are revealed.展开更多
With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the (2+1)-dimensional dispersive long wave equations are constructed by means of a function transfor...With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the (2+1)-dimensional dispersive long wave equations are constructed by means of a function transformation, improved mapping approach, and variable separation approach, among which there are rational solitary wave solutions, periodic wave solutions and rational wave solutions.展开更多
In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for (2+1)-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurca...In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for (2+1)-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.展开更多
Based on the computerized symbolic Maple, we study two important nonlinear evolution equations, i.e., the Hirota equation and the (1+1)-dimensional dispersive long wave equation by use of a direct and unified algebrai...Based on the computerized symbolic Maple, we study two important nonlinear evolution equations, i.e., the Hirota equation and the (1+1)-dimensional dispersive long wave equation by use of a direct and unified algebraic method named the general projective Riccati equation method to find more exact solutions to nonlinear differential equations. The method is more powerful than most of the existing tanh method. New and more general form solutions are obtained. The properties of the new formal solitary wave solutions are shown by some figures.展开更多
In this paper, a further extended Jacobi elliptic function rationM expansion method is proposed for constructing new forms of exact solutions to nonlinear partial differential equations by making a more general transf...In this paper, a further extended Jacobi elliptic function rationM expansion method is proposed for constructing new forms of exact solutions to nonlinear partial differential equations by making a more general transformation. For illustration, we apply the method to (2+1)-dimensionM dispersive long wave equation and successfully obtain many new doubly periodic solutions. When the modulus m→1, these sohitions degenerate as soliton solutions. The method can be also applied to other nonlinear partial differential equations.展开更多
With the help of a modified mapping method, we obtain two kinds of variable separation solutions with two arbitrary functions for the (24-1)-dimensional dispersive long wave equation. When selecting appropriate mult...With the help of a modified mapping method, we obtain two kinds of variable separation solutions with two arbitrary functions for the (24-1)-dimensional dispersive long wave equation. When selecting appropriate multi-valued functions in the variable separation solution, we investigate the interactions among special multi-dromions, dromion-like multi-peakons, and dromion-like multi-semifoldons, which all demonstrate non-completely elastic properties.展开更多
After generalizing the Clarkson-Kruskal direct similarity reduction ansatz, one can obtain various newtypes of reduction equations. Especially, some lower-dimensional turbulent systems or chaotic systems may be obtain...After generalizing the Clarkson-Kruskal direct similarity reduction ansatz, one can obtain various newtypes of reduction equations. Especially, some lower-dimensional turbulent systems or chaotic systems may be obtainedfrom the general form of the similarity reductions of a higher-dimensional Lax integrable model. Furthermore, anarbitrary three-order quasi-linear equation, which includes the Korteweg de-Vries Burgers equation and the generalLorenz equation as two special cases, has been obtained from the reductions of the (2+1)-dimensional dispersive longwave equation system. Some types of periodic and chaotic solutions of the system are also discussed.展开更多
The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the...The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The (2+1)-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.展开更多
This work is devoted to get a new family of analytical solutions of the(2+1)-coupled dispersive long wave equations propagating in an infinitely long channel with constant depth,and can be observed in an open sea or i...This work is devoted to get a new family of analytical solutions of the(2+1)-coupled dispersive long wave equations propagating in an infinitely long channel with constant depth,and can be observed in an open sea or in wide channels.The solutions are obtained by using the invariance property of the similarity transformations method via one-parameter Lie group theory.The repeated use of the similarity transformations method can transform the system of PDEs into system of ODEs.Under adequate restrictions,the reduced system of ODEs is solved.Numerical simulation is performed to describe the solutions in a physically meaningful way.The profiles of the solutions are simulated by taking an appropriate choice of functions and constants involved therein.In each animation,a frame for dominated behavior is captured.They exhibit elastic multisolitons,single soliton,doubly solitons,stationary,kink and parabolic nature.The results are significant since these have confirmed some of the established results of S.Kumar et al.(2020)and K.Sharma et al.(2020).Some of their solutions can be deduced from the results derived in this work.Other results in the existing literature are different from those in this work.展开更多
In the present paper, a simple and direct method was proposed to solve the (2+ 1)-dimensional long dispersive wave equations. A variable-dependent transformation was intorducedto convert the equations into the simpler...In the present paper, a simple and direct method was proposed to solve the (2+ 1)-dimensional long dispersive wave equations. A variable-dependent transformation was intorducedto convert the equations into the simpler forms, which are coupled and linear partial differentialequations, then obtain its general solution. Some special types of the localized excitations, suchas oscillating dromion, multi-solitoff, multi-dromion, multi-lump and multi-ring soliton solutionsare derived by selecting the arbitrary functions appropriately.展开更多
基金the Natural Science Foundation of Henan Province(0611050500)
文摘This paper concerns with the Cauchy problem for the nonlinear double dispersive wave equation. By the priori estimates and the method in [9], It proves that the Cauchy problem admits a unique global classical solution. And by the concave method, we give sufficient conditions on the blowup of the global solution for the Cauchy problem.
基金Supported by the National Natural Science Foundation of China the Doctoral Foundation of NEM of China
文摘In this note, we are concerned with the global singularity structures of weak solutions to 4 - D semilinear dispersive wave equations whose initial data are chosen to be singular at a single point, Combining Strichartz's inequality with the commutator argument techniques, we show that the weak solutions stay globally conormal if the Cauchy data are conormal
基金supported by the National Natural Science Foundation of China(10862003,40564001)the Innovative Research Team Building Programs of Inner Mongolia University for Nationalities
文摘The inverse problem for harmonic waves and wave packets was studied based on a full dispersive wave equation. First, a full dispersive wave equation which describes wave propagation in nondissipative microstructured linear solids is established based on the Mindlin theory, and the dispersion characteristics are discussed. Second, based on the full dispersive wave equation, an inverse problem for determining the four unknown coefficients of wave equa- tion is posed in terms of the frequencies and corresponding wave numbers of four different harmonic waves, and the inverse problem is demonstrated with rigorous mathematical theory. Research proves that the coefficients of wave equation related to material properties can be uniquely determined in cases of normal and anomalous dispersions by measuring the frequen- cies and corresponding wave numbers of four different harmonic waves which propagate in a nondissipative microstructured linear solids.
文摘A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a (1+1)-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11361069 and 11775146).
文摘We investigate (2+1)-dimensional generalized modified dispersive water wave (GMDWW) equation by utilizing the bifurcation theory of dynamical systems. We give the phase portraits and bifurcation analysis of the plane system corresponding to the GMDWW equation. By using the special orbits in the phase portraits, we analyze the existence of the traveling wave solutions. When some parameter takes special values, we obtain abundant exact kink wave solutions, singular wave solutions, periodic wave solutions, periodic singular wave solutions, and solitary wave solutions for the GMDWW equation.
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 Natural Science Foundation of Education Committee of Henan Province(2003110003)
文摘By using the homogeneous balance principle(HBP), we derive a Backtund transformation(BT) to the generalized dispersive long wave equation with variable coefficients.Based on the BT, we give many kinds of the exact solutions of the equation, such as, singlesolitary solutions, multi-soliton solutions and generalized exact solutions.
基金The project supported by China Postdoctoral Science Foundation, Natural Science Foundation of Zhejiang Province of China under Grant No. Y604056, and Doctor Foundation of Ningbo City under Grant No. 2005A610030
文摘By means of two different Riccati equations with different parameters as subequation in the components of finite rational expansion method, new complexiton solutions for the (1+1)-dimensional dispersive long wave equation are successfully constructed, which include various combination of trigonometric periodic and hyperbolic function solutions, various combination of trigonometric periodic and rational function solutions, and various combination of hyperbolic and rational function solutions.
基金The project supported in part by National Natural Science Foundation of China under Grant No. 10272071 and the Science Research Foundation of Huzhou University under Grant No. KX21025
文摘A new generalized extended F-expansion method is presented for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. As an application of this method, we study the (2+1)-dimensional dispersive long wave equation. With the aid of computerized symbolic computation, a number of doubly periodic wave solutions expressed by various Jacobi elliptic functions are obtained. In the limit cases, the solitary wave solutions are derived as well.
基金The project supported by National Natural Science Foundation of China under Grant No.10101025
文摘In this article, we study the Lax pairs of -dimensional equation: the modified generalized dispersive long wave (MGDLW) equation. Based on the well-known binary Darboux transformation, we dig out the recursion formulas of the first part of the Lax pairs. Then by further discussion and doing some revisional work, we make the recursion formulas fit for the second part of Lax pairs. At last, some solutions to the MGDLW equation are worked out by using the recursion formula.
基金The project supported by the China Postdoctoral Science Foundation under Grant No. 2004036086, K.C. Wong Education Foundation, Hong Kong, and partially supported by the State Key Basic Research Program of China under Grant No. 2004CB318000 . The authors are grateful to professor Gao Xiao-Shan for his enthusiastic guidance and help.
文摘Seeking exact analytical solutions of nonlinear evolution equations is of fundamental importance in mathematlcal physics. In this paper, based on a constructive algorithm and symbolic computation, abundant new exact solutions of the (2+1)-dimensional dispersive long wave equations are obtained, among which there are soliton-like solutions, mult-soliton-like solutions and formal periodic solutions, etc. Certain special solutions are considered and some interesting localized structures are revealed.
基金supported by the Scientific Research Foundation of Beijing Information Science and Technology UniversityScientific Creative Platform Foundation of Beijing Municipal Commission of Education
文摘With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the (2+1)-dimensional dispersive long wave equations are constructed by means of a function transformation, improved mapping approach, and variable separation approach, among which there are rational solitary wave solutions, periodic wave solutions and rational wave solutions.
基金Supported by the National Natural Science Foundation of China (10871206)Program for Excellent Talents in Guangxi Higher Education Institutions
文摘In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for (2+1)-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.
基金The project supported by National Natural Science Foundation of China under Grant No.10072013the State Key Basic Research Development Program under Grant No.G1998030600
文摘Based on the computerized symbolic Maple, we study two important nonlinear evolution equations, i.e., the Hirota equation and the (1+1)-dimensional dispersive long wave equation by use of a direct and unified algebraic method named the general projective Riccati equation method to find more exact solutions to nonlinear differential equations. The method is more powerful than most of the existing tanh method. New and more general form solutions are obtained. The properties of the new formal solitary wave solutions are shown by some figures.
基金The project partially supported by the State Key Basic Research Program of China under Grant No. 2004 CB 318000
文摘In this paper, a further extended Jacobi elliptic function rationM expansion method is proposed for constructing new forms of exact solutions to nonlinear partial differential equations by making a more general transformation. For illustration, we apply the method to (2+1)-dimensionM dispersive long wave equation and successfully obtain many new doubly periodic solutions. When the modulus m→1, these sohitions degenerate as soliton solutions. The method can be also applied to other nonlinear partial differential equations.
基金Supported by the National Natural Science Foundation of China (Grant No. 11005092)the Program for Innovative Research Team of Young Teachers,China (Grant No. 2009RC01)+1 种基金the Undergraduate Innovative Base of Zhejiang A & F University,Chinathe Zhejiang Province Undergraduate Scientific and Technological Innovation Project,China (Grant No. 2012R412018)
文摘With the help of a modified mapping method, we obtain two kinds of variable separation solutions with two arbitrary functions for the (24-1)-dimensional dispersive long wave equation. When selecting appropriate multi-valued functions in the variable separation solution, we investigate the interactions among special multi-dromions, dromion-like multi-peakons, and dromion-like multi-semifoldons, which all demonstrate non-completely elastic properties.
基金国家杰出青年科学基金,the Research Fund for Doctoral Program of HigherEducation of China,国家自然科学基金
文摘After generalizing the Clarkson-Kruskal direct similarity reduction ansatz, one can obtain various newtypes of reduction equations. Especially, some lower-dimensional turbulent systems or chaotic systems may be obtainedfrom the general form of the similarity reductions of a higher-dimensional Lax integrable model. Furthermore, anarbitrary three-order quasi-linear equation, which includes the Korteweg de-Vries Burgers equation and the generalLorenz equation as two special cases, has been obtained from the reductions of the (2+1)-dimensional dispersive longwave equation system. Some types of periodic and chaotic solutions of the system are also discussed.
文摘The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The (2+1)-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.
文摘This work is devoted to get a new family of analytical solutions of the(2+1)-coupled dispersive long wave equations propagating in an infinitely long channel with constant depth,and can be observed in an open sea or in wide channels.The solutions are obtained by using the invariance property of the similarity transformations method via one-parameter Lie group theory.The repeated use of the similarity transformations method can transform the system of PDEs into system of ODEs.Under adequate restrictions,the reduced system of ODEs is solved.Numerical simulation is performed to describe the solutions in a physically meaningful way.The profiles of the solutions are simulated by taking an appropriate choice of functions and constants involved therein.In each animation,a frame for dominated behavior is captured.They exhibit elastic multisolitons,single soliton,doubly solitons,stationary,kink and parabolic nature.The results are significant since these have confirmed some of the established results of S.Kumar et al.(2020)and K.Sharma et al.(2020).Some of their solutions can be deduced from the results derived in this work.Other results in the existing literature are different from those in this work.
文摘In the present paper, a simple and direct method was proposed to solve the (2+ 1)-dimensional long dispersive wave equations. A variable-dependent transformation was intorducedto convert the equations into the simpler forms, which are coupled and linear partial differentialequations, then obtain its general solution. Some special types of the localized excitations, suchas oscillating dromion, multi-solitoff, multi-dromion, multi-lump and multi-ring soliton solutionsare derived by selecting the arbitrary functions appropriately.