This paper considers the generalized KdV equation with or without natural boundary conditions and provides a parameter region for solitons and solitary waves, and also modifies a result of Zabuskys. The solitary bifur...This paper considers the generalized KdV equation with or without natural boundary conditions and provides a parameter region for solitons and solitary waves, and also modifies a result of Zabuskys. The solitary bifurcation has been discussed.展开更多
By using the theory of compensated compactness,we prove that there exists a sequence {uδε} converges nearly everywhere to the solution of the initial-value problem of generalized KdV equation with high order perturb...By using the theory of compensated compactness,we prove that there exists a sequence {uδε} converges nearly everywhere to the solution of the initial-value problem of generalized KdV equation with high order perturbation terms,namely we prove the existence of the weak solution.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 means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painleve test for integrability only for three distin...By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painleve test for integrability only for three distinct cases. Moreover, the multi- soliton solutions are presented for this equation under three sets of integrable conditions. Finally, by selecting appropriate parameters, we analyze the evolution of two solitons, which is especially interesting as it may describe the overtaking and the head-on collisions of solitary waves of different shapes and different types.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
The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this articl...The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics. The linear superposition solutions to the generalized KdV equation K（2,2,1）, the Oliver water wave equation, and the k（n, n） equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed.展开更多
The dynamical behaviour of the generalized Korteweg-de Vries （KdV） equation under a periodic perturbation is investigated numerically. The bifurcation and chaos in the system are observed by applying bifurcation dia...The dynamical behaviour of the generalized Korteweg-de Vries （KdV） equation under a periodic perturbation is investigated numerically. The bifurcation and chaos in the system are observed by applying bifurcation diagrams, phase portraits and Poincar＇e maps. To characterise the chaotic behaviour of this system, the spectra of the Lyapunov exponent and Lyapunov dimension of the attractor are also employed.展开更多
In this study,the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV)describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability prope...In this study,the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV)describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained.Moreover,several new solutions have been constructed for the gvcmKdV.Additionally,the classical direct similarity reduction method is used to re-duce the gvcmKdV to a nonlinear ordinary differential equation.Building on the solutions given in the previous literature for the reduced equation,many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.展开更多
By using a weakly nonlinear and perturbation method, the generalized inhomogeneous Korteweg de Vries （KdV）- Burgers equation is derived, which governs the evolution of the amplitude of Rossby waves under the influen...By using a weakly nonlinear and perturbation method, the generalized inhomogeneous Korteweg de Vries （KdV）- Burgers equation is derived, which governs the evolution of the amplitude of Rossby waves under the influence of dissipation and slowly varying topography with time. The analysis indicates that dissipation and slowly varying topography with time are important factors in causing variation in the mass and energy of solitary waves.展开更多
基金Research partially supported by Shanghai Development Grant of Education Committee(# 2 0 0 0 A1 0 )
文摘This paper considers the generalized KdV equation with or without natural boundary conditions and provides a parameter region for solitons and solitary waves, and also modifies a result of Zabuskys. The solitary bifurcation has been discussed.
基金Supported by the Innovation Talents of Science and Technology of Henan University(2009-HASTIT-007)Supported by the Natural Science Program of Department of Education(2011A110006)
文摘By using the theory of compensated compactness,we prove that there exists a sequence {uδε} converges nearly everywhere to the solution of the initial-value problem of generalized KdV equation with high order perturbation terms,namely we prove the existence of the weak solution.
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Project supported by the National Natural Science Foundation of China (Grant Nos. 11201290 and 71103118)
文摘By means of singularity structure analysis, the integrability of a generalized fifth-order KdV equation is investigated. It is proven that this equation passes the Painleve test for integrability only for three distinct cases. Moreover, the multi- soliton solutions are presented for this equation under three sets of integrable conditions. Finally, by selecting appropriate parameters, we analyze the evolution of two solitons, which is especially interesting as it may describe the overtaking and the head-on collisions of solitary waves of different shapes and different types.
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The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics. The linear superposition solutions to the generalized KdV equation K（2,2,1）, the Oliver water wave equation, and the k（n, n） equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10875078)the Natural Science Foundation of Zhejiang Province,China (Grant No. Y7080455)
文摘The dynamical behaviour of the generalized Korteweg-de Vries （KdV） equation under a periodic perturbation is investigated numerically. The bifurcation and chaos in the system are observed by applying bifurcation diagrams, phase portraits and Poincar＇e maps. To characterise the chaotic behaviour of this system, the spectra of the Lyapunov exponent and Lyapunov dimension of the attractor are also employed.
基金The author would like to thank the Deanship of Scientific Re-search,Majmaah University,Saudi Arabia,for funding this work under project No.R-2021-222.
文摘In this study,the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV)describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained.Moreover,several new solutions have been constructed for the gvcmKdV.Additionally,the classical direct similarity reduction method is used to re-duce the gvcmKdV to a nonlinear ordinary differential equation.Building on the solutions given in the previous literature for the reduced equation,many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.
基金supported by the Knowledge Innovation Key Program of the Chinese Academy of Sciences (Grant No. KZCX1-YW-12)the National Key Science Foundation of China (Grant No. 41030855)
文摘By using a weakly nonlinear and perturbation method, the generalized inhomogeneous Korteweg de Vries （KdV）- Burgers equation is derived, which governs the evolution of the amplitude of Rossby waves under the influence of dissipation and slowly varying topography with time. The analysis indicates that dissipation and slowly varying topography with time are important factors in causing variation in the mass and energy of solitary waves.
