In this paper, we consider the generalized Korteweg-de-Vries (KdV) equations which are remarkable models of the water waves mechanics, the shallow water waves, the quantum mechanics, the ion acoustic waves in plasma, ...In this paper, we consider the generalized Korteweg-de-Vries (KdV) equations which are remarkable models of the water waves mechanics, the shallow water waves, the quantum mechanics, the ion acoustic waves in plasma, the electro-hydro-dynamical model for local electric field, signal processing waves through optical fibers, etc. We determine the useful and further general exact traveling wave solutions of the above mentioned NLDEs by applying the exp(−τ(ξ))-expansion method by aid of traveling wave transformations. Furthermore, we explain the physical significance of the obtained solutions of its definite values of the involved parameters with graphic representations in order to know the physical phenomena. Finally, we show that the exp(−τ(ξ))-expansion method is convenient, powerful, straightforward and provide more general solutions and can be helping to examine vast amount of travelling wave solutions to the other different kinds of NLDEs.展开更多
The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensivel...The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensively provided.Then,we give parametric expressions of different types of solutions matching with the corresponding orbits.Finally,solution profiles,3D and density plots of some solutions are presented with proper parametric choices.展开更多
By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of ...By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg de Vries (KdV) equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.展开更多
This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equation and a coupled modified Korteweg-de Vries (mKdV)...This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equation and a coupled modified Korteweg-de Vries (mKdV) equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.展开更多
By using reductive perturbation method, super KdV equations are changed into ordinary KdV equations, small amplitude perturbation solutions are obtained.
In this work, we study the generalized Rosenau-KdV equation. We shall use the sech-ansatze method to derive the solitary wave solutions of this equation.
A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and th...A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of hell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.展开更多
This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete mu...This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect.展开更多
By asing the nonclassical method of symmetry reductions, the exact solutions for general variable coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and non...By asing the nonclassical method of symmetry reductions, the exact solutions for general variable coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and nonuniformity terms don't exist, the multisoliton solutions are found and the corresponding Painleve II type equation for the variable coefficient KdV equation is given.展开更多
This paper applies an extended auxiliary equation method to obtain exact solutions of the KdV equation with variable coefficients. As a result, solitary wave solutions, trigonometric function solutions, rational funct...This paper applies an extended auxiliary equation method to obtain exact solutions of the KdV equation with variable coefficients. As a result, solitary wave solutions, trigonometric function solutions, rational function solutions, Jacobi elliptic doubly periodic wave solutions, and nonsymmetrical kink solution are obtained. It is shown that the extended auxiliary equation method, with the help of a computer symbolic computation system, is reliable and effective in finding exact solutions of variable coefficient nonlinear evolution equations in mathematical physics.展开更多
This article is concerned with the Hirota direct method for studying novel multisoliton solutions of the discrete KdV equation. First the Hirota method was introduced, then the novel multisoliton solutions were obtain...This article is concerned with the Hirota direct method for studying novel multisoliton solutions of the discrete KdV equation. First the Hirota method was introduced, then the novel multisoliton solutions were obtained. Simultaneously the figures of the novel one-soliton solution and two-soliton solution were given and the singularity of the novel multisoliton solutions was discussed. Finally it was pointed out that the multisoliton solutions with sigularity can only be called soliton-like solutions. Key words differential-difference KdV equation - Hirota method - multisoliton-like solutions MSC 2000 35Q51 Project supported by the National Natural Science Foundation of China(Grant No. 19571052)展开更多
The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutio...The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutions (including kinky periodic solitary-wave solutions, periodic soliton solutions, and cross kink-wave solutions) for the new (2+1)-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensionai nonlinear wave field.展开更多
The simple Lie point symmetry reduction procedure is used to obtain infinitely many symmetries to a new integrable system of coupled KdV equations. Using some symmetry subalgebra of the equations, five types of the si...The simple Lie point symmetry reduction procedure is used to obtain infinitely many symmetries to a new integrable system of coupled KdV equations. Using some symmetry subalgebra of the equations, five types of the significant similarity reductions are obtained by virtue of the Lie group approach, and obtain abundant solutions of the coupled KdV equations, such as the solitary wave solution, exponential solution, rational solution, polynomial solution, etc.展开更多
The Korteweg-de Vries equation with a forcing term is established by recent studies as a simple mathematicalmodel of describing the physics of a shallow layer of fluid subject to external forcing.In the present paper,...The Korteweg-de Vries equation with a forcing term is established by recent studies as a simple mathematicalmodel of describing the physics of a shallow layer of fluid subject to external forcing.In the present paper,we study theanalytic solutions to the KdV equation with forcing term by using Hirota's direct method.Several exact solutions aregiven as examples,from which one can see that the same type soliton solutions can be excited by different forced term.展开更多
In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the (1+1)-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which...In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the (1+1)-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.展开更多
To further study the nal solitons on the continental fission laws of initial intershelf/slope, we rederive and correct the 2D KdV equation of Djordjevic & Redekopp for exponentially stratified fluid (or ocean) and ...To further study the nal solitons on the continental fission laws of initial intershelf/slope, we rederive and correct the 2D KdV equation of Djordjevic & Redekopp for exponentially stratified fluid (or ocean) and with twodimensional topography. Through a combination of theoretical study and numerical experiments, we show that solitons in the odd vertical modes can fission. However, because of the corrections, the fission conditions are different from those of Djordjevic & Redekopp. The even modes cannot fission unless the initial internal solitons propagate from shallow sea to deep sea. This conclusion is entirely opposite to that of Djordjevic & Redekopp.展开更多
The coupled Korteweg-de Vries (CKdV) equation with self-consistent sources (CKdVESCS) and its Lax representation are derived. We present a generalized binary Darboux transformation (GBDT) with an arbitrary time-...The coupled Korteweg-de Vries (CKdV) equation with self-consistent sources (CKdVESCS) and its Lax representation are derived. We present a generalized binary Darboux transformation (GBDT) with an arbitrary time- dependent function for the CKdVESCS as well as the formula for the N-times repeated GBDT. This GBDT provides non-auto-Biicklund transformation between two CKdVESCSs with different degrees of sources and enables us to construct more generM solutions with N arbitrary t-dependent functions. We obtain positon, negaton, complexiton, and negaton- positon solutions of the CKdVESCS.展开更多
Based on the closed connections among the homogeneous balance (HB) method and Clarkson-KruSkal (CK) method, we study the similarity reductions of the generalized variable coefficients 2D KdV equation. In the meant...Based on the closed connections among the homogeneous balance (HB) method and Clarkson-KruSkal (CK) method, we study the similarity reductions of the generalized variable coefficients 2D KdV equation. In the meantime it is shown that this leads to a direct reduction in the form of ordinary differential equation under some integrability conditions between the variable coefficients. Two different cases have been discussed, the search for solutions of those ordinary differential equations yielded many exact travelling and solitonic wave solutions in the form of hyperbolic and trigonometric functions under some constraints between the variable coefficients.展开更多
In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the...In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travelling wave form satisfies some special conditions.展开更多
In this paper we consider exact solutions to the KdV equation under the Bargmann constraint. Solutions expressed through exponential polynomials and Wronskians are derived by bilinear approach through solving the Lax ...In this paper we consider exact solutions to the KdV equation under the Bargmann constraint. Solutions expressed through exponential polynomials and Wronskians are derived by bilinear approach through solving the Lax pair under the Bargmann constraint. It is also shown that the potential u in the stationary Sehrodinger equation can be a summation of squares of wave functions from bilinear point of view.展开更多
文摘In this paper, we consider the generalized Korteweg-de-Vries (KdV) equations which are remarkable models of the water waves mechanics, the shallow water waves, the quantum mechanics, the ion acoustic waves in plasma, the electro-hydro-dynamical model for local electric field, signal processing waves through optical fibers, etc. We determine the useful and further general exact traveling wave solutions of the above mentioned NLDEs by applying the exp(−τ(ξ))-expansion method by aid of traveling wave transformations. Furthermore, we explain the physical significance of the obtained solutions of its definite values of the involved parameters with graphic representations in order to know the physical phenomena. Finally, we show that the exp(−τ(ξ))-expansion method is convenient, powerful, straightforward and provide more general solutions and can be helping to examine vast amount of travelling wave solutions to the other different kinds of NLDEs.
基金Project supported by the Natural Science Foundation of Shandong Province (Grant No.ZR2021MA084)the Natural Science Foundation of Liaocheng University (Grant No.318012025)Discipline with Strong Characteristics of Liaocheng University–Intelligent Science and Technology (Grant No.319462208)。
文摘The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensively provided.Then,we give parametric expressions of different types of solutions matching with the corresponding orbits.Finally,solution profiles,3D and density plots of some solutions are presented with proper parametric choices.
基金Project supported by the National Natural Science Foundation of China(Grant No 10461006), the High Education Science Research Program(Grant No NJ02035) of Inner Mongolia Autonomous Region, Natural Science Foundation of Inner Mongolia Autonomous Region(Grant No 2004080201103) and the Youth Research Program of Inner Mongolia Normal University(Grant No QN005023).
文摘By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg de Vries (KdV) equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10771019 and 10826107)
文摘This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries (KdV) equation and a coupled modified Korteweg-de Vries (mKdV) equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.
文摘By using reductive perturbation method, super KdV equations are changed into ordinary KdV equations, small amplitude perturbation solutions are obtained.
文摘In this work, we study the generalized Rosenau-KdV equation. We shall use the sech-ansatze method to derive the solitary wave solutions of this equation.
基金*The project supported by National Natural Science Foundation of China under Grant No. 10471139 and Hong Kong Research Grant Council under Grant No. HKBU/2016/03P
文摘A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of hell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10572119, 10772147 and 10632030)the Doctoral Program Foundation of Education Ministry of China (Grant No 20070699028)+1 种基金the National Natural Science Foundation of Shaanxi Province of China (Grant No 2006A07)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment
文摘This paper considers the multi-symplectic formulations of the generalized fifth-order KdV equation in Hamiltonian space. Recurring to the midpoint rule, it presents an implicit multi-symplectic scheme with discrete multi-symplectic conservation law to solve the partial differential equations which are derived from the generalized fifth-order KdV equation numerically. The results of the numerical experiments show that this multi-symplectic algorithm is good in accuracy and its long-time numerical behaviour is also perfect.
基金Supported by the Develop Programme Foundation of the National Basic research(G1 9990 3 2 80 1 )
文摘By asing the nonclassical method of symmetry reductions, the exact solutions for general variable coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and nonuniformity terms don't exist, the multisoliton solutions are found and the corresponding Painleve II type equation for the variable coefficient KdV equation is given.
基金Project supported by the Fundamental Research Funds for the Central Universities (Grant No. 2010B17914) and the National Natural Science Foundation of China (Grant No. 10926162).
文摘This paper applies an extended auxiliary equation method to obtain exact solutions of the KdV equation with variable coefficients. As a result, solitary wave solutions, trigonometric function solutions, rational function solutions, Jacobi elliptic doubly periodic wave solutions, and nonsymmetrical kink solution are obtained. It is shown that the extended auxiliary equation method, with the help of a computer symbolic computation system, is reliable and effective in finding exact solutions of variable coefficient nonlinear evolution equations in mathematical physics.
文摘This article is concerned with the Hirota direct method for studying novel multisoliton solutions of the discrete KdV equation. First the Hirota method was introduced, then the novel multisoliton solutions were obtained. Simultaneously the figures of the novel one-soliton solution and two-soliton solution were given and the singularity of the novel multisoliton solutions was discussed. Finally it was pointed out that the multisoliton solutions with sigularity can only be called soliton-like solutions. Key words differential-difference KdV equation - Hirota method - multisoliton-like solutions MSC 2000 35Q51 Project supported by the National Natural Science Foundation of China(Grant No. 19571052)
基金Supported by the Natural Science Foundation of China under Grant Nos.10361007,10661002Yunnan Natural Science Foundation under Grant No.2006A0082M
文摘The extended homoclinic test function method is a kind of classic, efficient and well-developed method to solve nonlinear evolution equations. In this paper, with the help of this approach, we obtain new exact solutions (including kinky periodic solitary-wave solutions, periodic soliton solutions, and cross kink-wave solutions) for the new (2+1)-dimensional KdV equation. These results enrich the variety of the dynamics of higher-dimensionai nonlinear wave field.
基金The project supported by National Natural Science Foundation of China under Grant No. 10071033 and the Natural Science Foundation of Jiangsu Province under Grant No. BK2002003. Acknowledgments 0ne of the authors (S.P. Qian) is indebted to Prof. S.Y. Lou for his helpful discussions.
文摘The simple Lie point symmetry reduction procedure is used to obtain infinitely many symmetries to a new integrable system of coupled KdV equations. Using some symmetry subalgebra of the equations, five types of the significant similarity reductions are obtained by virtue of the Lie group approach, and obtain abundant solutions of the coupled KdV equations, such as the solitary wave solution, exponential solution, rational solution, polynomial solution, etc.
基金Supported by the GUCAS President Grant,the National Natural Science Foundation of China under Grant No.10701076
文摘The Korteweg-de Vries equation with a forcing term is established by recent studies as a simple mathematicalmodel of describing the physics of a shallow layer of fluid subject to external forcing.In the present paper,we study theanalytic solutions to the KdV equation with forcing term by using Hirota's direct method.Several exact solutions aregiven as examples,from which one can see that the same type soliton solutions can be excited by different forced term.
基金supported by the National Natural Science Foundation of China (No.10461005)the Ph.D.Programs Foundation of Ministry of Education of China (No.20070128001)the High Education Science Research Program of Inner Mongolia (No.NJZY08057)
文摘In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the (1+1)-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.
基金The project supported by the National Natural Science Foundation of China(40276008)the Grant of Key Laboratory of Marine Science and Numerical Modeling.SOA(0201(2003))
文摘To further study the nal solitons on the continental fission laws of initial intershelf/slope, we rederive and correct the 2D KdV equation of Djordjevic & Redekopp for exponentially stratified fluid (or ocean) and with twodimensional topography. Through a combination of theoretical study and numerical experiments, we show that solitons in the odd vertical modes can fission. However, because of the corrections, the fission conditions are different from those of Djordjevic & Redekopp. The even modes cannot fission unless the initial internal solitons propagate from shallow sea to deep sea. This conclusion is entirely opposite to that of Djordjevic & Redekopp.
基金The project supported by the National Fundamental Research Program of China(973 Program)under Grant No.2007CB814800National Natural Science Foundation of China under Grant No.10601028
文摘The coupled Korteweg-de Vries (CKdV) equation with self-consistent sources (CKdVESCS) and its Lax representation are derived. We present a generalized binary Darboux transformation (GBDT) with an arbitrary time- dependent function for the CKdVESCS as well as the formula for the N-times repeated GBDT. This GBDT provides non-auto-Biicklund transformation between two CKdVESCSs with different degrees of sources and enables us to construct more generM solutions with N arbitrary t-dependent functions. We obtain positon, negaton, complexiton, and negaton- positon solutions of the CKdVESCS.
文摘Based on the closed connections among the homogeneous balance (HB) method and Clarkson-KruSkal (CK) method, we study the similarity reductions of the generalized variable coefficients 2D KdV equation. In the meantime it is shown that this leads to a direct reduction in the form of ordinary differential equation under some integrability conditions between the variable coefficients. Two different cases have been discussed, the search for solutions of those ordinary differential equations yielded many exact travelling and solitonic wave solutions in the form of hyperbolic and trigonometric functions under some constraints between the variable coefficients.
基金The project supported by National Natural Science Foundation of China under Grant No. 10371023 and Shanghai Leading Academic Discipline Project under Grant No. T0502)
文摘In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travelling wave form satisfies some special conditions.
基金Supported by National Natural Science Foundation of China under Grant Nos. 10871165 and 10671121
文摘In this paper we consider exact solutions to the KdV equation under the Bargmann constraint. Solutions expressed through exponential polynomials and Wronskians are derived by bilinear approach through solving the Lax pair under the Bargmann constraint. It is also shown that the potential u in the stationary Sehrodinger equation can be a summation of squares of wave functions from bilinear point of view.