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Adequate Closed Form Wave Solutions to the Generalized KdV Equation in Mathematical Physics
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作者 Md. Munnu Miah Md. Al Amin Meia +1 位作者 Md. Matiur Rahman Sarker Ahammodullah Hasan 《Journal of Applied Mathematics and Physics》 2024年第6期2069-2082,共14页
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
关键词 The Generalized kdv equation The exp(-τ(ξ)) -Expansion Method Travelling Wave Solitary Wave
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Exact solutions of a time-fractional modified KdV equation via bifurcation analysis
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作者 刘敏远 许慧 王增桂 《Chinese Physics B》 SCIE EI CAS CSCD 2023年第12期192-199,共8页
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
关键词 the time-fractional modified kdv equation bifurcation analysis exact solutions
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The Jacobi elliptic function-like exact solutions to two kinds of KdV equations with variable coefficients and KdV equation with forcible term 被引量:10
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作者 套格图桑 斯仁到尔吉 《Chinese Physics B》 SCIE EI CAS CSCD 2006年第12期2809-2818,共10页
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
关键词 auxiliary equation kdv equation with variable coefficients kdv equation with a forcible term Jacobi elliptic function-like exact solutions
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Approximate analytic solutions for a generalized Hirota—Satsuma coupled KdV equation and a coupled mKdV equation
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作者 赵国忠 蔚喜军 +2 位作者 徐云 朱江 吴迪 《Chinese Physics B》 SCIE EI CAS CSCD 2010年第8期46-54,共9页
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
关键词 approximate analytic solutions generalized Hirota-Satsuma coupled kdv equation coupled mkdv equation variational iteration method
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REDUCTIVE PERTURBATION METHOD OF SUPER KdV EQUATIONS
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作者 吕克璞 孙建安 +1 位作者 段文山 赵金保 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2001年第7期846-850,共5页
By using reductive perturbation method, super KdV equations are changed into ordinary KdV equations, small amplitude perturbation solutions are obtained.
关键词 super kdv equation reductive perturbation method kdv equation
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Solitary Wave Solutions for Generalized Rosenau-KdV Equation 被引量:11
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作者 Amin Esfahani 《Communications in Theoretical Physics》 SCIE CAS CSCD 2011年第3期396-398,共3页
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.
关键词 SOLITONS Ansatze method Rosenau equation kdv equation
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Higher-Dimensional KdV Equations and Their Soliton Solutions 被引量:12
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作者 ZHANG Yu-Feng Tam Honwah ZHAO Jing 《Communications in Theoretical Physics》 SCIE CAS CSCD 2006年第3期411-413,共3页
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
关键词 bilinear operator kdv equation soliton equation
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Multi-symplectic method for generalized fifth-order KdV equation 被引量:6
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作者 胡伟鹏 邓子辰 《Chinese Physics B》 SCIE EI CAS CSCD 2008年第11期3923-3929,共7页
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
关键词 generalized fifth-order kdv equation MULTI-SYMPLECTIC travelling wave solution conservation law
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EXACT SOLUTIONS FOR GENERAL VARIABLE-COEFFICIENT KdV EQUATION 被引量:8
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作者 Liu Xiqiang Jiang SongGraduate School, China Academy of Engineering and Physics, P.O. Box 2101, Beijing 100088 Dept. of Math., Liaocheng Teachers Univ., Shandong 252000. Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beiji 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2001年第4期377-380,共4页
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
关键词 General variable coefficient kdv equation nonclassical method of symmetry reduction exact solution.
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The extended auxiliary the KdV equation with equation method for variable coefficients 被引量:8
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作者 Shi Lan-Fang Chen Cai-Sheng Zhou Xian-Chun 《Chinese Physics B》 SCIE EI CAS CSCD 2011年第10期166-170,共5页
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
关键词 extended auxiliary equation method kdv equation with variable coefficients exactsolutions
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Novel Multisoliton-Like Solutions of the Differential-Difference KdV Equation 被引量:7
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作者 杜丛民 邓淑芳 孙梅娜 《Journal of Shanghai University(English Edition)》 CAS 2004年第2期134-137,共4页
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
关键词 differential-difference kdv equation Hirota method multisoliton-like solutions
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New Exact Periodic Solitary-Wave Solutions for New (2+1)-Dimensional KdV Equation 被引量:4
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作者 WANG Chuan-Jia DAI Zheng-De MU Gui LIN Song-Qing 《Communications in Theoretical Physics》 SCIE CAS CSCD 2009年第11期862-864,共3页
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
关键词 kinky wave solution periodic soliton solution extended homoclinic test function method kdv equation bilinear form
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Lie Point Symmetries and Exact Solutions of Couple KdV Equations 被引量:5
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作者 QIAN Su-Ping TIAN Li-Xin 《Communications in Theoretical Physics》 SCIE CAS CSCD 2007年第4期582-586,共5页
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
关键词 coupled kdv equations Lie point symmetry exact solutions
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Analytic Solutions to Forced KdV Equation 被引量:3
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作者 ZHAO Jun-Xiao GUO Bo-Ling 《Communications in Theoretical Physics》 SCIE CAS CSCD 2009年第8期279-283,共5页
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
关键词 the forced kdv equation SOLITONS bilinear method TSUNAMI
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A new auxiliary equation method for finding travelling wave solutions to KdV equation 被引量:3
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作者 庞晶 边春泉 朝鲁 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2010年第7期929-936,共8页
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
关键词 auxiliary equation method travelling wave solution kdv equation homogeneous balance method
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A correction of the 2D KdV equation of Djordjevic & Redekopp in exponentially stratified fluid 被引量:2
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作者 Zhaoting Xu Guojin Shen Samuel Shan-pu Shen 《Acta Mechanica Sinica》 SCIE EI CAS CSCD 2005年第4期346-352,共7页
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
关键词 Internal wave SOLITON FISSION 2D kdv equation
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On Coupled KdV Equations with Self-consistent Sources 被引量:2
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作者 HUANG Ye-Hui WU Hong-Xia +1 位作者 XIE Xi ZENG Yun-Bo 《Communications in Theoretical Physics》 SCIE CAS CSCD 2008年第5期1091-1100,共10页
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
关键词 coupled kdv equation with self-consistent sources generalized binary Darboux transformation POSITON NEGATON COMPLEXITON
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Direct Reduction and Exact Solutions for Generalized Variable Coefficients 2D KdV Equation under Some Integrability Conditions 被引量:2
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作者 M.H.M.Moussa RehabM.El-Shiekh 《Communications in Theoretical Physics》 SCIE CAS CSCD 2011年第4期551-554,共4页
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
关键词 direct reduction method the generalized variable coefficients 2D kdv equation exact solutions
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Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation 被引量:2
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作者 ZHANG Wei-Guo DONG Chun-Yan FAN En-Gui 《Communications in Theoretical Physics》 SCIE CAS CSCD 2006年第6X期1091-1100,共10页
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
关键词 generalized compound kdv equation generalized compound kdv-Burgers equation
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Solving the KdV Equation Under Bargmann Constraint via Bilinear Approach 被引量:1
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作者 张建兵 张大军 陈登远 《Communications in Theoretical Physics》 SCIE CAS CSCD 2010年第2期211-217,共7页
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
关键词 the kdv equation the Bargmann constraint Lax pair N-soliton solution
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