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
In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions ar...In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation（s）.展开更多
By using a reconstruction procedure of conservation laws of different models,the deformation algorithm proposed by Lou,Hao and Jia has been used to a new application such that a decoupled system becomes a coupled one....By using a reconstruction procedure of conservation laws of different models,the deformation algorithm proposed by Lou,Hao and Jia has been used to a new application such that a decoupled system becomes a coupled one.Using the new application to some decoupled systems such as the decoupled dispersionless Korteweg–de Vries(Kd V)systems related to dispersionless waves,the decoupled KdV systems related to dispersion waves,the decoupled KdV and Burgers systems related to the linear dispersion and diffusion effects,and the decoupled KdV and Harry–Dym(HD)systems related to the linear and nonlinear dispersion effects,we have obtained various new types of higher dimensional integrable coupled systems.The new models can be used to describe the interactions among different nonlinear waves and/or different effects including the dispersionless waves(dispersionless KdV waves),the linear dispersion waves(KdV waves),the nonlinear dispersion waves(HD waves)and the diffusion effect.The method can be applied to couple all different separated integrable models.展开更多
Kortweg-de Vries （KdV）-typed equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Generalized complex coupled KdV （GCCKdV） equations are investigated in this paper. Through the d...Kortweg-de Vries （KdV）-typed equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Generalized complex coupled KdV （GCCKdV） equations are investigated in this paper. Through the dependent variable transformations and symbolic computation, GCCKdV equations are transformed into their bilinear forms, based on which the one- and two-soliton solutions are obtained. Through the interactions of two solitons, the regular elastic collision are shown. When the wave numbers are complex, three kinds of solitonie collisions are presented： （i） two solitons merge and separate from each other periodically; （ii） two solitons exhibit the attraction and repulsion nearly twice, and finally separate from each other after such type of interaction; （iii） two solitons are ftuctuant in the central region of the collision. Propagation features of solitons are investigated with the effects of the coefficients in the GCCKdV equations considered. Velocity of soliton increase with the a increasing. Amplitude of v increase with the a increasing and decrease with the β increasing.展开更多
In this article, we construct the exact traveling wave solutions for nonlinear evolution equations in the mathematical physics via the modified Kawahara equation, the nonlinear coupled KdV equations and the classical ...In this article, we construct the exact traveling wave solutions for nonlinear evolution equations in the mathematical physics via the modified Kawahara equation, the nonlinear coupled KdV equations and the classical Boussinesq equations, by using a generalized （G＇/G）-expansion method, where G satisfies the Jacobi elliptic equation. Many exact solutions in terms of Jacobi elliptic functions are obtained.展开更多
We develop the generalized conditional symmetry （GCS） approach to solve the problem of dimensional reduction of Cauchy problems for the KdV-type equations. We characterize these equations that admit certain higheror...We develop the generalized conditional symmetry （GCS） approach to solve the problem of dimensional reduction of Cauchy problems for the KdV-type equations. We characterize these equations that admit certain higherorder GCSs and show the main reduction procedure by some examples. The obtained reductions cannot be derived within the framework of the standard Lie approach.展开更多
Amplitude equations governing the nonlinear resonant interaction of equatorial baroclinic and barotropic Rossby waves were derived by Majda and Biello and used as a model for long range interactions （teleconnections...Amplitude equations governing the nonlinear resonant interaction of equatorial baroclinic and barotropic Rossby waves were derived by Majda and Biello and used as a model for long range interactions （teleconnections） between the tropical and midlatitude troposphere. An overview of that derivation is nonlinear wave theory, but not in atmospheric presented and geared to readers versed in sciences. In the course of the derivation, two other sets of asymptotic equations are presented： the long equatorial wave equations and the weakly nonlinear, long equatorial wave equations. A linear transformation recasts the amplitude equations as nonlinear and linearly coupled KdV equations governing the amplitude of two types of modes, each of which consists of a coupled tropical/midlatitude flow. In the limit of Rossby waves with equal dispersion, the transformed amplitude equations become two KdV equations coupled only through nonlinear fluxes. Four numerical integrations are presented which show （i） the interaction of two solitons, one from either mode, （ii） and （iii） the interaction of a soliton in the presence of different mean wind shears, and （iv） the interaction of two solitons mediated by the presence of a mean wind shear.展开更多
Special solution of the （2＋1）-dimensional Sawada Kotera equation is decomposed into three （0＋1）- dimensional Bargmann flows. They are straightened out on the Jacobi variety of the associated hyperelliptic curve....Special solution of the （2＋1）-dimensional Sawada Kotera equation is decomposed into three （0＋1）- dimensional Bargmann flows. They are straightened out on the Jacobi variety of the associated hyperelliptic curve. Explicit algebraic-geometric solution is obtained on the basis of a deeper understanding of the KdV hierarchy.展开更多
Based on the invariant expansion method, some reasonable approximate solutions of coupled Korteweg-de Vries （KdV） equations with different linear dispersion relations have been obtained. These solutions contain not ...Based on the invariant expansion method, some reasonable approximate solutions of coupled Korteweg-de Vries （KdV） equations with different linear dispersion relations have been obtained. These solutions contain not only bell type soliton solutions but also periodic wave solutions that expressed by Jacob/elliptic functions. The results also show that if the arbitrary constants are selected suitably, the approximate solutions may become the exact ones.展开更多
Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamilto...Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differentiM equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable ease of the two-dimensional Hamiltonian system.展开更多
This paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili(KdV-KP)equation.This equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Pe...This paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili(KdV-KP)equation.This equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Petviashvili equation.The newly gathered class of sixth-order KdV-KP equation is studied using the sub-equation method to obtain several soliton-type solutions which consist of trigonometric,hyperbolic,and rational solutions.The application of the sub-equation approach in this work draws attention to the outstanding characteristics of the suggested method and its ability to handle completely integrable equations.Furthermore,the obtained solutions have not been reported in the previous literature and might have significant impact on future research.展开更多
A universal in tailing wave-train generation of forced soliton generationover topography is found theoretically as the flows are at the resonant points and it is examinedwith the numerical calculation of the correspon...A universal in tailing wave-train generation of forced soliton generationover topography is found theoretically as the flows are at the resonant points and it is examinedwith the numerical calculation of the corresponding fKdV e-quatioa From the comparisons, it is shownthat theoretical and numerical results on the invariance is in good agreement and the theory givenin this paper does not include the modulus truncation, any free constant and unknown function.展开更多
Using the direct method for a coupled KdV system, six types of the similarity reductions are obtained. The group explanation of the results is also given. It is pointed out that, in order to find all the results by no...Using the direct method for a coupled KdV system, six types of the similarity reductions are obtained. The group explanation of the results is also given. It is pointed out that, in order to find all the results by nonclassical Lie approach, two additional condition equations should be satisfied at the same time together with two original equations.展开更多
In this paper,several arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)methods are presented for Korteweg-de Vries(KdV)type equations on moving meshes.Based on the L^(2) conservation law of KdV equations,we...In this paper,several arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)methods are presented for Korteweg-de Vries(KdV)type equations on moving meshes.Based on the L^(2) conservation law of KdV equations,we adopt the conservative and dissipative numerical fuxes for the nonlinear convection and linear dispersive terms,respectively.Thus,one conservative and three dissipative ALE-DG schemes are proposed for the equations.The invariant preserving property for the conservative scheme and the corresponding dissipative properties for the other three dissipative schemes are all presented and proved in this paper.In addition,the L^(2)-norm error estimates are also proved for two schemes,whose numerical fuxes for the linear dispersive term are both dissipative type.More precisely,when choosing the approximation space with the piecewise kth degree polynomials,the error estimate provides the kth order of convergence rate in L^(2)-norm for the scheme with the conservative numerical fuxes applied for the nonlinear convection term.Furthermore,the(k+1∕2)th order of accuracy can be proved for the ALE-DG scheme with dissipative numerical fuxes applied for the convection term.Moreover,a Hamiltonian conservative ALE-DG scheme is also presented based on the conservation of the Hamiltonian for KdV equations.Numerical examples are shown to demonstrate the accuracy and capability of the moving mesh ALE-DG methods and compare with stationary DG methods.展开更多
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The project supported by the State Key Basic Research Program of China under Grant No 2004CB318000
文摘In this paper, we present a method to solve difference differential equation（s）. As an example, we apply this method to discrete KdV equation and Ablowitz-Ladik lattice equation. As a result, many exact solutions are obtained with the help of Maple including soliton solutions presented by hyperbolic functions sinh and cosh, periodic solutions presented by sin and cos and rational solutions. This method can also be used to other nonlinear difference-differential equation（s）.
基金The National Natural Science Foundation(Nos.12235007,12090020,11975131,12090025)。
文摘By using a reconstruction procedure of conservation laws of different models,the deformation algorithm proposed by Lou,Hao and Jia has been used to a new application such that a decoupled system becomes a coupled one.Using the new application to some decoupled systems such as the decoupled dispersionless Korteweg–de Vries(Kd V)systems related to dispersionless waves,the decoupled KdV systems related to dispersion waves,the decoupled KdV and Burgers systems related to the linear dispersion and diffusion effects,and the decoupled KdV and Harry–Dym(HD)systems related to the linear and nonlinear dispersion effects,we have obtained various new types of higher dimensional integrable coupled systems.The new models can be used to describe the interactions among different nonlinear waves and/or different effects including the dispersionless waves(dispersionless KdV waves),the linear dispersion waves(KdV waves),the nonlinear dispersion waves(HD waves)and the diffusion effect.The method can be applied to couple all different separated integrable models.
基金＊Supported by the National Natural Science Foundation of China under Grant No. 60772023, by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. SKLSDE-07-001, Beijing University of Aeronautics and Astronautics, by the National Basic Research Program of China （973 Program） under Grant No. 2005CB321901, and by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos. 20060006024 and 200800130006, Chinese Ministry of Education.
文摘Kortweg-de Vries （KdV）-typed equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Generalized complex coupled KdV （GCCKdV） equations are investigated in this paper. Through the dependent variable transformations and symbolic computation, GCCKdV equations are transformed into their bilinear forms, based on which the one- and two-soliton solutions are obtained. Through the interactions of two solitons, the regular elastic collision are shown. When the wave numbers are complex, three kinds of solitonie collisions are presented： （i） two solitons merge and separate from each other periodically; （ii） two solitons exhibit the attraction and repulsion nearly twice, and finally separate from each other after such type of interaction; （iii） two solitons are ftuctuant in the central region of the collision. Propagation features of solitons are investigated with the effects of the coefficients in the GCCKdV equations considered. Velocity of soliton increase with the a increasing. Amplitude of v increase with the a increasing and decrease with the β increasing.
文摘In this article, we construct the exact traveling wave solutions for nonlinear evolution equations in the mathematical physics via the modified Kawahara equation, the nonlinear coupled KdV equations and the classical Boussinesq equations, by using a generalized （G＇/G）-expansion method, where G satisfies the Jacobi elliptic equation. Many exact solutions in terms of Jacobi elliptic functions are obtained.
基金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 develop the generalized conditional symmetry （GCS） approach to solve the problem of dimensional reduction of Cauchy problems for the KdV-type equations. We characterize these equations that admit certain higherorder GCSs and show the main reduction procedure by some examples. The obtained reductions cannot be derived within the framework of the standard Lie approach.
基金Project supported by the National Science Foundation (No.DMS-0604947)
文摘Amplitude equations governing the nonlinear resonant interaction of equatorial baroclinic and barotropic Rossby waves were derived by Majda and Biello and used as a model for long range interactions （teleconnections） between the tropical and midlatitude troposphere. An overview of that derivation is nonlinear wave theory, but not in atmospheric presented and geared to readers versed in sciences. In the course of the derivation, two other sets of asymptotic equations are presented： the long equatorial wave equations and the weakly nonlinear, long equatorial wave equations. A linear transformation recasts the amplitude equations as nonlinear and linearly coupled KdV equations governing the amplitude of two types of modes, each of which consists of a coupled tropical/midlatitude flow. In the limit of Rossby waves with equal dispersion, the transformed amplitude equations become two KdV equations coupled only through nonlinear fluxes. Four numerical integrations are presented which show （i） the interaction of two solitons, one from either mode, （ii） and （iii） the interaction of a soliton in the presence of different mean wind shears, and （iv） the interaction of two solitons mediated by the presence of a mean wind shear.
基金The project supported by the Special Funds for Major State Basic Research Project under Grant No.G2000077301
文摘Special solution of the （2＋1）-dimensional Sawada Kotera equation is decomposed into three （0＋1）- dimensional Bargmann flows. They are straightened out on the Jacobi variety of the associated hyperelliptic curve. Explicit algebraic-geometric solution is obtained on the basis of a deeper understanding of the KdV hierarchy.
基金Supported by National Natural Science Foundation of China under Grant No.11104248Zhejiang Provincial Natural Science Foundation of China under Grant No.LQ12A01008Project of Education of Zhejiang Province under Grant No.Y201327716
文摘Based on the invariant expansion method, some reasonable approximate solutions of coupled Korteweg-de Vries （KdV） equations with different linear dispersion relations have been obtained. These solutions contain not only bell type soliton solutions but also periodic wave solutions that expressed by Jacob/elliptic functions. The results also show that if the arbitrary constants are selected suitably, the approximate solutions may become the exact ones.
基金the Funds for Basic Research Project under Grant Nos.06XJC033 and 2008Bl10003
文摘Based on the second integrable ease of known two-dimensional Hamiltonian system with a quartie potentiM, we propose a 4 × 4 matrix speetrM problem and derive a hierarchy of coupled KdV equations and their Hamiltonian structures. It is shown that solutions of the coupled KdV equations in the hierarchy are reduced to solving two compatible systems of ordinary differentiM equations. As an application, quite a few explicit solutions of the coupled KdV equations are obtained via using separability for the second integrable ease of the two-dimensional Hamiltonian system.
文摘This paper studied new class of integral equation called the Korteweg-de Vries-Kadomtsev-Petviashvili(KdV-KP)equation.This equation consist of the well-known fifth-order KdV equation in the context of the Kadomtsev-Petviashvili equation.The newly gathered class of sixth-order KdV-KP equation is studied using the sub-equation method to obtain several soliton-type solutions which consist of trigonometric,hyperbolic,and rational solutions.The application of the sub-equation approach in this work draws attention to the outstanding characteristics of the suggested method and its ability to handle completely integrable equations.Furthermore,the obtained solutions have not been reported in the previous literature and might have significant impact on future research.
文摘A universal in tailing wave-train generation of forced soliton generationover topography is found theoretically as the flows are at the resonant points and it is examinedwith the numerical calculation of the corresponding fKdV e-quatioa From the comparisons, it is shownthat theoretical and numerical results on the invariance is in good agreement and the theory givenin this paper does not include the modulus truncation, any free constant and unknown function.
基金The project supported by National Natural Science Foundation of China under Grant No. 10071033, the Natural Science Foundation of Jiangsu Province under Grant No. BK2002003, the Project of Technology Innovation Plan for Postgraduate of Jiangsu Province in Year 2006 under Grant No. 72, and the Natural Science Directed Foundation of the Jiangsu Higher Education Institutions under Grant No. 06KJDll0001
文摘Using the direct method for a coupled KdV system, six types of the similarity reductions are obtained. The group explanation of the results is also given. It is pointed out that, in order to find all the results by nonclassical Lie approach, two additional condition equations should be satisfied at the same time together with two original equations.
基金This work was supported by the National Numerical Windtunnel Project NNW2019ZT4-B08Science Challenge Project TZZT2019-A2.3the National Natural Science Foundation of China Grant no.11871449.
文摘In this paper,several arbitrary Lagrangian-Eulerian discontinuous Galerkin(ALE-DG)methods are presented for Korteweg-de Vries(KdV)type equations on moving meshes.Based on the L^(2) conservation law of KdV equations,we adopt the conservative and dissipative numerical fuxes for the nonlinear convection and linear dispersive terms,respectively.Thus,one conservative and three dissipative ALE-DG schemes are proposed for the equations.The invariant preserving property for the conservative scheme and the corresponding dissipative properties for the other three dissipative schemes are all presented and proved in this paper.In addition,the L^(2)-norm error estimates are also proved for two schemes,whose numerical fuxes for the linear dispersive term are both dissipative type.More precisely,when choosing the approximation space with the piecewise kth degree polynomials,the error estimate provides the kth order of convergence rate in L^(2)-norm for the scheme with the conservative numerical fuxes applied for the nonlinear convection term.Furthermore,the(k+1∕2)th order of accuracy can be proved for the ALE-DG scheme with dissipative numerical fuxes applied for the convection term.Moreover,a Hamiltonian conservative ALE-DG scheme is also presented based on the conservation of the Hamiltonian for KdV equations.Numerical examples are shown to demonstrate the accuracy and capability of the moving mesh ALE-DG methods and compare with stationary DG methods.
