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
With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a res...With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a result, their abundant new soliton-like solutions and period form solutions are found.展开更多
With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna- Pemp...With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna- Pempinelli equation. Then, the corresponding figures of the periodic wave solutions are given. Fhrthermore, it is shown that the known soliton solutions can be reduced from the periodic wave solutions.展开更多
The exact chirped soliton-like and quasi-periodic wave solutions of （2 ＋ 1）-dimensional generalized nonlinear Schr6dinger equation including linear and nonlinear gain （loss） with variable coefficients are obtaine...The exact chirped soliton-like and quasi-periodic wave solutions of （2 ＋ 1）-dimensional generalized nonlinear Schr6dinger equation including linear and nonlinear gain （loss） with variable coefficients are obtained detalledly in this paper. The form and the behavior of solutions are strongly affected by the modulation of both the dispersion coefficient and the nonlinearity coefficient. In addition, self-similar soliton-like waves precisely piloted from our obtained solutions by tailoring the dispersion and linear gain （loss）.展开更多
In the present letter, we get the appropriate bilinear forms of （2 ＋ 1）-dimensional KdV equation, extended （2 ＋ 1）-dimensional shallow water wave equation and （2 ＋ 1）-dimensional Sawada -Kotera equation in a ...In the present letter, we get the appropriate bilinear forms of （2 ＋ 1）-dimensional KdV equation, extended （2 ＋ 1）-dimensional shallow water wave equation and （2 ＋ 1）-dimensional Sawada -Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.展开更多
By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified wa...By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m → 1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.展开更多
The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown th...The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown that more novel kinds of solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form,and so on.展开更多
This paper applies the EXP-function method to find exact solutions of various nonlinear equations. Tzitzeica- Dodd-Bullough (TDB) equation was selected to illustrate the effectiveness and convenience of the suggested ...This paper applies the EXP-function method to find exact solutions of various nonlinear equations. Tzitzeica- Dodd-Bullough (TDB) equation was selected to illustrate the effectiveness and convenience of the suggested method. More generalized solitonary solutions with free parameters were obtained by suitable choice of the free parameters, and also the obtained solitonary solutions can be converted into periodic solutions.展开更多
In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for two coupled nonlinear partial differential equations are obtained.
Exact heteroclinic breather-wave solutions for Davey-Stewartson （DSI, DSII） system with periodic boundary condition are constructed using Hirota＇s bilinear form method and generalized ansatz method. The heteroclini...Exact heteroclinic breather-wave solutions for Davey-Stewartson （DSI, DSII） system with periodic boundary condition are constructed using Hirota＇s bilinear form method and generalized ansatz method. The heteroclinic structure of wave is investigated.展开更多
By constructing appropriate transformations and an extended elliptic sub-equation approach, we find some exact solutions of variable coefficient cubic-quintie nonlinear Schrodinger equation with an external potential,...By constructing appropriate transformations and an extended elliptic sub-equation approach, we find some exact solutions of variable coefficient cubic-quintie nonlinear Schrodinger equation with an external potential, which include bell and kink profile solitary wave solutions, singular solutions, triangular periodic wave solutions and so on.展开更多
The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations.By using the bifurcation theory of dynamical system...The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations.By using the bifurcation theory of dynamical systems to do qualitative analysis,all possible phase portraits in the parametric space for the traveling wave systems are obtained.It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied.The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.展开更多
By employing Hirota bilinear method and Riemann theta functions of genus one,explicit triply periodic wave solutions for the(2+1)-dimensional Boussinesq equation are constructed under the Backlund transformation u =(1...By employing Hirota bilinear method and Riemann theta functions of genus one,explicit triply periodic wave solutions for the(2+1)-dimensional Boussinesq equation are constructed under the Backlund transformation u =(1 /6)(u0 1) + 2[ln f(x,y,t)] xx,four kinds of triply periodic wave solutions are derived,and their long wave limit are discussed.The properties of one of the solutions are shown in Fig.1.展开更多
We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schr6dinger equation with timeand space-dependent distributed coefficients in harmonic and optical lattice potenti...We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schr6dinger equation with timeand space-dependent distributed coefficients in harmonic and optical lattice potentials. We utilize the similarity transformation technique to obtain these solutions. Constraints for the dispersion coefficient, the nonlinearity, and the gain （loss） coefficient are presented at the same time. Various shapes of periodic wave and soliton solutions are studied analytically and physically. Stability analysis of the solutions is discussed numerically.展开更多
The global dynamics of a cholera model with delay is considered. We determine a basic reproduction number R0 which is chosen based on the relative ODE model, and establish that the global dynamics are determined by th...The global dynamics of a cholera model with delay is considered. We determine a basic reproduction number R0 which is chosen based on the relative ODE model, and establish that the global dynamics are determined by the threshold value R0. If R0 〈 1, then the infection-free equilibrium is global asymptotically stable, that is, the cholera dies out; If R0 〉 1, then the unique endemic equilibrium is global asymptotically stable, which means that the infection persists. The results obtained show that the delay does not lead to periodic oscillations. Finally, some numerical simulations support our theoretical results.展开更多
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 the National Key Basic Research Development Project Program under Grant No.G1998030600the Foundation of Liaoning Normal University
文摘With the use of computer algebra, the method that straightforwardly leads to travelling wave solutions is presented. The compound KdV-Burgers equation and KP-B equation are chosen to illustrate this approach. As a result, their abundant new soliton-like solutions and period form solutions are found.
基金Supported by the National Natural Science Foundation of China under Grant No.60772023the Open Fund of the State Key Laboratory of Software Development Environment under Grant No.BUAA-SKLSDE-09KF-04+1 种基金Beijing University of Aeronautics and Astronautics,by the National Basic Research Program of China (973 Program) under Grant No.2005CB321901the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos.20060006024 and 200800130006,Chinese Ministry of Education
文摘With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna- Pempinelli equation. Then, the corresponding figures of the periodic wave solutions are given. Fhrthermore, it is shown that the known soliton solutions can be reduced from the periodic wave solutions.
基金Supported by the National Natural Science Foundation of China under Grant No.11072219the Zhejiang Provincial Natural Science Foundation under Grant No.Y1100099
文摘The exact chirped soliton-like and quasi-periodic wave solutions of （2 ＋ 1）-dimensional generalized nonlinear Schr6dinger equation including linear and nonlinear gain （loss） with variable coefficients are obtained detalledly in this paper. The form and the behavior of solutions are strongly affected by the modulation of both the dispersion coefficient and the nonlinearity coefficient. In addition, self-similar soliton-like waves precisely piloted from our obtained solutions by tailoring the dispersion and linear gain （loss）.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11075055,61021004,10735030Shanghai Leading Academic Discipline Project under Grant No.B412Program for Changjiang Scholars and Innovative Research Team in University(IRT0734)
文摘In the present letter, we get the appropriate bilinear forms of （2 ＋ 1）-dimensional KdV equation, extended （2 ＋ 1）-dimensional shallow water wave equation and （2 ＋ 1）-dimensional Sawada -Kotera equation in a quick and natural manner, namely by appling the binary Bell polynomials. Then the Hirota direct method and Riemann theta function are combined to construct the periodic wave solutions of the three types nonlinear evolution equations. And the corresponding figures of the periodic wave solutions are given. Furthermore, the asymptotic properties of the periodic wave solutions indicate that the soliton solutions can be derived from the periodic wave solutions.
文摘By using the generally projective Riccati equation method, a series of doubly periodic solutions (Jacobi elliptic function solution) for a class of nonlinear partial differential equations are obtained in a unified way. When the module m → 1, these solutions exactly degenerate to the soliton solutions of the equations. Then we reveal the relationship between the soliton-like solutions obtained by other authors and these soliton solutions of the equations.
文摘The new rational form solutions to the elliptic equation are shown, and then these solutions to the elliptic equation are taken as a transformation and applied to solve nonlinear coupled wave equations. It is shown that more novel kinds of solutions are derived, such as periodic solutions of rational form, solitary wave solutions of rational form,and so on.
文摘This paper applies the EXP-function method to find exact solutions of various nonlinear equations. Tzitzeica- Dodd-Bullough (TDB) equation was selected to illustrate the effectiveness and convenience of the suggested method. More generalized solitonary solutions with free parameters were obtained by suitable choice of the free parameters, and also the obtained solitonary solutions can be converted into periodic solutions.
基金The project supported by National Natural Science Foundation of China under Grant Nos. 90511009 and 40305006 Cprrespondence author,
文摘In this paper, by applying the Jacobi elliptic function expansion method, the periodic solutions for two coupled nonlinear partial differential equations are obtained.
基金Supported by Chinese Natural Science Foundation under Grant No. 10661002Yunnan Natural Science Foundation under Grant No. 2006A0082M
文摘Exact heteroclinic breather-wave solutions for Davey-Stewartson （DSI, DSII） system with periodic boundary condition are constructed using Hirota＇s bilinear form method and generalized ansatz method. The heteroclinic structure of wave is investigated.
基金supported by National Natural Science Foundation of China under Grant No.10172056
文摘By constructing appropriate transformations and an extended elliptic sub-equation approach, we find some exact solutions of variable coefficient cubic-quintie nonlinear Schrodinger equation with an external potential, which include bell and kink profile solitary wave solutions, singular solutions, triangular periodic wave solutions and so on.
基金National Natural Science Foundation of China(No.19731003,No.19961003)Yunnan Provincial Natural Science Foundation of China(No.1999A0018M,No.2000A0002M)
文摘The method of the phase plane is emploied to investigate the solitary and periodic traveling waves for a class of nonlinear dispersive partial differential equations.By using the bifurcation theory of dynamical systems to do qualitative analysis,all possible phase portraits in the parametric space for the traveling wave systems are obtained.It can be shown that the existence of a singular straight line in the traveling wave system is the reason why smooth solitary wave solutions converge to solitary cusp wave solution when parameters are varied.The different parameter conditions for the existence of solitary and periodic wave solutions of different kinds are rigorously determined.
基金Supported by the National Natural Science Foundation of China under Grant No. 11101382the Natural Science Foundation of Henan Province under Grant No. 2010A110001the Basic and Advanced Technology Project of Henan Province under Grant No. 112300410199
文摘By employing Hirota bilinear method and Riemann theta functions of genus one,explicit triply periodic wave solutions for the(2+1)-dimensional Boussinesq equation are constructed under the Backlund transformation u =(1 /6)(u0 1) + 2[ln f(x,y,t)] xx,four kinds of triply periodic wave solutions are derived,and their long wave limit are discussed.The properties of one of the solutions are shown in Fig.1.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10875106 and 11175158
文摘We construct analytical periodic wave and soliton solutions to the generalized nonautonomous nonlinear Schr6dinger equation with timeand space-dependent distributed coefficients in harmonic and optical lattice potentials. We utilize the similarity transformation technique to obtain these solutions. Constraints for the dispersion coefficient, the nonlinearity, and the gain （loss） coefficient are presented at the same time. Various shapes of periodic wave and soliton solutions are studied analytically and physically. Stability analysis of the solutions is discussed numerically.
文摘The global dynamics of a cholera model with delay is considered. We determine a basic reproduction number R0 which is chosen based on the relative ODE model, and establish that the global dynamics are determined by the threshold value R0. If R0 〈 1, then the infection-free equilibrium is global asymptotically stable, that is, the cholera dies out; If R0 〉 1, then the unique endemic equilibrium is global asymptotically stable, which means that the infection persists. The results obtained show that the delay does not lead to periodic oscillations. Finally, some numerical simulations support our theoretical results.
