In this paper, based on hyperbolic tanh-function method and homogeneous balance method, and auxiliary equation method, some new exact solitary solutions to the generalized mKdV equation and generalized Zakharov-Kuzent...In this paper, based on hyperbolic tanh-function method and homogeneous balance method, and auxiliary equation method, some new exact solitary solutions to the generalized mKdV equation and generalized Zakharov-Kuzentsov equation are constructed by the method of auxiliary equation with function transformation with aid of symbolic computation system Mathematica. The method is of important significance in seeking new exact solutions to the evolution equation with arbitrary nonlinear term.展开更多
This paper uses the weakly nonlinear method and perturbation method to deal with the quasi-geostrophic vorticity equation,and the modified Korteweg-de Vries（mKdV） equations describing the evolution of the amplitude ...This paper uses the weakly nonlinear method and perturbation method to deal with the quasi-geostrophic vorticity equation,and the modified Korteweg-de Vries（mKdV） equations describing the evolution of the amplitude of solitary Rossby waves as the change of Rossby parameter β（у） with latitude у is obtained.展开更多
In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified...In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified Benjamin-Bona- Mahoney （mBBM） equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions： amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional deriva- tives are described in the modified Riemann-Liouville sense.展开更多
We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and the...We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.展开更多
Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the com...Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg-de Vries （KdV） and modified KdV （mKdV） equation and （2＋1）-dimensional Broer-Kaup-Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.展开更多
Hirota＇s bilinear direct method is applied to constructing soliton solutions to a special coupled modified Korteweg- de Vries （mKdV） system. Some physical properties such as the spatiotemporal evolution, waveform s...Hirota＇s bilinear direct method is applied to constructing soliton solutions to a special coupled modified Korteweg- de Vries （mKdV） system. Some physical properties such as the spatiotemporal evolution, waveform structure, interactive phenomena of solitons are discussed, especially in the two-soliton case. It is found that different interactive behaviours of solitary waves take place under different parameter conditions of overtaking collision in this system. It is verified that the elastic interaction phenomena exist in this （1＋1）-dimensional integrable coupled model.展开更多
Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equat...Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations. New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney （BBM） equations are obtained with the aid of computer algebraic system Maple. The method is also valid for other （l＋l）-dimensional and higher dimensional systems.展开更多
We present a new reliable analytical study for solving the discontinued problems arising in nanotechnology. Such problems are presented as nonlinear differential-difference equations. The proposed method is based on t...We present a new reliable analytical study for solving the discontinued problems arising in nanotechnology. Such problems are presented as nonlinear differential-difference equations. The proposed method is based on the Laplace trans- form with the homotopy analysis method （HAM）. This method is a powerful tool for solving a large amount of problems. This technique provides a series of functions which may converge to the exact solution of the problem. A good agreement between the obtained solution and some well-known results is obtained.展开更多
A quite general coupled variable coefficient modified KdV (VCmKdV) equation in a two-layer fluid systemis derived by means of the reductive perturbation method.Making use of the CK's direct method,some similarityr...A quite general coupled variable coefficient modified KdV (VCmKdV) equation in a two-layer fluid systemis derived by means of the reductive perturbation method.Making use of the CK's direct method,some similarityreductions of the coupled VCmKdV equation are obtained and their corresponding group explanations are discussed.Some exact solutions of the coupled equations are also presented.展开更多
Recently the （G′/G）-expansion method was proposed to find the traveling wave solutions of nonlinear evolution equations. This paper shows that the （G′/G）-expansion method is a special form of the truncated Pain...Recently the （G′/G）-expansion method was proposed to find the traveling wave solutions of nonlinear evolution equations. This paper shows that the （G′/G）-expansion method is a special form of the truncated Painlev＇e expansion method by introducing an intermediate expansion method. Then the generalized （G′/G）-（G/G′） expansion method is naturally derived from the standpoint of the nonstandard truncated Painlev＇e expansion. The application of the generalized method to the mKdV equation shows that it extends the range of exact solutions obtained by using the （ G′/ G）-expansion method.展开更多
Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigatedsystematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcation...Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigatedsystematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcationparameter. Then, different conditions expressed in terms of the bifurcation parameter are obtained for the existence ofbreather-like, algebraic, pulse-like solitary waves, and shock waves. All types of the solitary wave and shock wave solutionsare given by direct integration. Finally, an approximate analytic method by employing the interpolation polynomials iscomplete and the theoretical methods are the simplest hitherto.展开更多
This paper gives a recursion operator for a 1-constrained CKP hierarchy, and by the recursion operator it proves that the 1-constrained CKP hierarchy can be reduced to the mKdV hierarchy under condition q = r.
In this work, by means of a new more general ansatz and the symbolic computation system Maple, we extend the Riccati equation rational expansion method [Chaos, Solitons ＆ Fractals 25 （2005） 1019] to uniformly const...In this work, by means of a new more general ansatz and the symbolic computation system Maple, we extend the Riccati equation rational expansion method [Chaos, Solitons ＆ Fractals 25 （2005） 1019] to uniformly construct a series of stochastic nontravelling wave solutions for nonlinear stochastic evolution equation. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions including a series of rational formal nontraveling wave and coefficient functions＇ soliton-like solution.s and trigonometric-like function solutions. The method can also be applied to solve other nonlinear stochastic evolution equation or equations.展开更多
N-soliton solutions of the hierarchy of non-isospectral mKdV equation with self-consistent sources andthe hierarchy of non-isospectral sine-Gordon equation with self-consistent sources are obtained via the inverse sca...N-soliton solutions of the hierarchy of non-isospectral mKdV equation with self-consistent sources andthe hierarchy of non-isospectral sine-Gordon equation with self-consistent sources are obtained via the inverse scatteringtransform.展开更多
In this letter, we study discretized mKdV lattice equation by using a new generalized ansatz. As a result,many explicit rational exact solutions, including some new solitary wave solutions, are obtained by symbolic co...In this letter, we study discretized mKdV lattice equation by using a new generalized ansatz. As a result,many explicit rational exact solutions, including some new solitary wave solutions, are obtained by symbolic computation code Maple.展开更多
Under consideration in this study is the discrete coupled modified Korteweg-de Vries(mKdV)equation with 4×4 Lax pair.Firstly,through using continuous limit technique,this discrete equation can be mapped to the co...Under consideration in this study is the discrete coupled modified Korteweg-de Vries(mKdV)equation with 4×4 Lax pair.Firstly,through using continuous limit technique,this discrete equation can be mapped to the coupled KdV and mKdV equations,which may depict the development of shallow water waves,the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma.Secondly,the discrete generalized(r,N-r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem,from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background,higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived,and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique.Finally,the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions.These results may be helpful for understanding some physical phenomena in fields of shallow water wave,optics,and plasma physics.展开更多
In this paper, an extended car-following model is proposed based on an optimal velocity model (OVM), which takes the speed limit into consideration. The model is analyzed by using the linear stability theory and nonli...In this paper, an extended car-following model is proposed based on an optimal velocity model (OVM), which takes the speed limit into consideration. The model is analyzed by using the linear stability theory and nonlinear analysis method. The linear stability condition shows that the speed limit can enlarge the stable region of traffic flow. By applying the reductive perturbation method, the time-dependent Ginzburg-Landau (TDGL) equation and the modified Korteweg-de Vries (mKdV) equation are derived to describe the traffic flow near the critical point. Furthermore, the relation between TDGL and mKdV equations is also given. It is clarified that the speed limit is essentially equivalent to the parameter adjusting of the driver’s sensitivity.展开更多
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In this paper, based on hyperbolic tanh-function method and homogeneous balance method, and auxiliary equation method, some new exact solitary solutions to the generalized mKdV equation and generalized Zakharov-Kuzentsov equation are constructed by the method of auxiliary equation with function transformation with aid of symbolic computation system Mathematica. The method is of important significance in seeking new exact solutions to the evolution equation with arbitrary nonlinear term.
基金Project supported by the Educational Department of Inner Mongolia (NJZY:08005)Open Fund of the Key Laboratory of Ocean Circulation and Waves,Chinese Academy of Sciences (Grant No KLOCAW0805)
文摘This paper uses the weakly nonlinear method and perturbation method to deal with the quasi-geostrophic vorticity equation,and the modified Korteweg-de Vries（mKdV） equations describing the evolution of the amplitude of solitary Rossby waves as the change of Rossby parameter β（у） with latitude у is obtained.
文摘In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified Benjamin-Bona- Mahoney （mBBM） equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions： amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional deriva- tives are described in the modified Riemann-Liouville sense.
文摘We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.
基金Project supported by the National Natural Science Foundation of China （Grant No 10472029）.
文摘Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg-de Vries （KdV） and modified KdV （mKdV） equation and （2＋1）-dimensional Broer-Kaup-Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.
文摘Hirota＇s bilinear direct method is applied to constructing soliton solutions to a special coupled modified Korteweg- de Vries （mKdV） system. Some physical properties such as the spatiotemporal evolution, waveform structure, interactive phenomena of solitons are discussed, especially in the two-soliton case. It is found that different interactive behaviours of solitary waves take place under different parameter conditions of overtaking collision in this system. It is verified that the elastic interaction phenomena exist in this （1＋1）-dimensional integrable coupled model.
基金Supported by the National Natural Science Foundation of China (No. 10647112)the Foundation of Donghua University
文摘Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations. New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney （BBM） equations are obtained with the aid of computer algebraic system Maple. The method is also valid for other （l＋l）-dimensional and higher dimensional systems.
文摘We present a new reliable analytical study for solving the discontinued problems arising in nanotechnology. Such problems are presented as nonlinear differential-difference equations. The proposed method is based on the Laplace trans- form with the homotopy analysis method （HAM）. This method is a powerful tool for solving a large amount of problems. This technique provides a series of functions which may converge to the exact solution of the problem. A good agreement between the obtained solution and some well-known results is obtained.
基金The project supported by National Natural Science Foundation of China under Grant Nos.10547124,10475055,and 90503006the Youth Foundation of Shanghai Jiao Tong University
文摘A quite general coupled variable coefficient modified KdV (VCmKdV) equation in a two-layer fluid systemis derived by means of the reductive perturbation method.Making use of the CK's direct method,some similarityreductions of the coupled VCmKdV equation are obtained and their corresponding group explanations are discussed.Some exact solutions of the coupled equations are also presented.
基金Project supported by the National Key Basic Research Project of China (Grant No. 2004CB318000)the National Natural Science Foundation of China (Grant No. 10771072)
文摘Recently the （G′/G）-expansion method was proposed to find the traveling wave solutions of nonlinear evolution equations. This paper shows that the （G′/G）-expansion method is a special form of the truncated Painlev＇e expansion method by introducing an intermediate expansion method. Then the generalized （G′/G）-（G/G′） expansion method is naturally derived from the standpoint of the nonstandard truncated Painlev＇e expansion. The application of the generalized method to the mKdV equation shows that it extends the range of exact solutions obtained by using the （ G′/ G）-expansion method.
文摘Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigatedsystematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcationparameter. Then, different conditions expressed in terms of the bifurcation parameter are obtained for the existence ofbreather-like, algebraic, pulse-like solitary waves, and shock waves. All types of the solitary wave and shock wave solutionsare given by direct integration. Finally, an approximate analytic method by employing the interpolation polynomials iscomplete and the theoretical methods are the simplest hitherto.
基金NSFC (10671187 10971109)the Program for NCET (NECT-08-0515)
文摘This paper gives a recursion operator for a 1-constrained CKP hierarchy, and by the recursion operator it proves that the 1-constrained CKP hierarchy can be reduced to the mKdV hierarchy under condition q = r.
基金The author would like to thank the referees very much for their careful reading of the manuscript and many valuable suggestions.
文摘In this work, by means of a new more general ansatz and the symbolic computation system Maple, we extend the Riccati equation rational expansion method [Chaos, Solitons ＆ Fractals 25 （2005） 1019] to uniformly construct a series of stochastic nontravelling wave solutions for nonlinear stochastic evolution equation. To illustrate the effectiveness of our method, we take the stochastic mKdV equation as an example, and successfully construct some new and more general solutions including a series of rational formal nontraveling wave and coefficient functions＇ soliton-like solution.s and trigonometric-like function solutions. The method can also be applied to solve other nonlinear stochastic evolution equation or equations.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10371070,10671121the Foundation of Shanghai Education Committee for Shanghai Prospective Excellent Young Teachers+1 种基金Shanghai Leading Academic Discipline Project under Grant No.J50101 the President Foundation of East China Institute of Technology under Grant No.DHXK0810
文摘N-soliton solutions of the hierarchy of non-isospectral mKdV equation with self-consistent sources andthe hierarchy of non-isospectral sine-Gordon equation with self-consistent sources are obtained via the inverse scatteringtransform.
基金the National Key Basic Research Project of China under
文摘In this letter, we study discretized mKdV lattice equation by using a new generalized ansatz. As a result,many explicit rational exact solutions, including some new solitary wave solutions, are obtained by symbolic computation code Maple.
基金Project supported by the National Natural Science Foundation of China (Grant No.12071042)Beijing Natural Science Foundation (Grant No.1202006)。
文摘Under consideration in this study is the discrete coupled modified Korteweg-de Vries(mKdV)equation with 4×4 Lax pair.Firstly,through using continuous limit technique,this discrete equation can be mapped to the coupled KdV and mKdV equations,which may depict the development of shallow water waves,the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma.Secondly,the discrete generalized(r,N-r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem,from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background,higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived,and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique.Finally,the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions.These results may be helpful for understanding some physical phenomena in fields of shallow water wave,optics,and plasma physics.
文摘In this paper, an extended car-following model is proposed based on an optimal velocity model (OVM), which takes the speed limit into consideration. The model is analyzed by using the linear stability theory and nonlinear analysis method. The linear stability condition shows that the speed limit can enlarge the stable region of traffic flow. By applying the reductive perturbation method, the time-dependent Ginzburg-Landau (TDGL) equation and the modified Korteweg-de Vries (mKdV) equation are derived to describe the traffic flow near the critical point. Furthermore, the relation between TDGL and mKdV equations is also given. It is clarified that the speed limit is essentially equivalent to the parameter adjusting of the driver’s sensitivity.
