This paper is concerned with the generalized nonlinear second-order equation.By the direct construction method,all of the first-order multipliers of the equation are obtained,and the corresponding complete conservatio...This paper is concerned with the generalized nonlinear second-order equation.By the direct construction method,all of the first-order multipliers of the equation are obtained,and the corresponding complete conservation laws(CLs) of such equations are provided.Furthermore,the integrability of the equation is considered in terms of the conservation laws.In addition,the relationship of multipliers and symmetries of the equations is investigated.展开更多
By using the theory of bifurcations of planar dynamic systems to the coupled Jaulent-Miodek equations, the existence of smooth solitary travelling wave solutions and uncountably infinite many smooth periodic travellin...By using the theory of bifurcations of planar dynamic systems to the coupled Jaulent-Miodek equations, the existence of smooth solitary travelling wave solutions and uncountably infinite many smooth periodic travelling wave solutions is studied and the bifurcation parametric sets are shown. Under the given parametric conditions, all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained.展开更多
By using the theory of the generalized perturbed Hamiltonian systems, it is shown that there exist periodic stream lines in the three_dimensional square cell pattern of Rayleigh_Benard convection. The result means tha...By using the theory of the generalized perturbed Hamiltonian systems, it is shown that there exist periodic stream lines in the three_dimensional square cell pattern of Rayleigh_Benard convection. The result means that our method enables this three_dimensional flow pattern to be described in an unambiguous manner, and some experimental results of other authors can be explained.展开更多
By using the theory of planar dynamical systems to the ion acoustic plasma equations, we obtain the existence of the solutions of the smooth and non-smooth solitary waves and the uncountably infinite smooth and non-sm...By using the theory of planar dynamical systems to the ion acoustic plasma equations, we obtain the existence of the solutions of the smooth and non-smooth solitary waves and the uncountably infinite smooth and non-smooth periodic waves. Under the given parametric conditions, we present the sufficient conditions to guarantee the existence of the above solutions.展开更多
Analytical techniques and Liapunov method were used for the estimation of the attraction domain of memory patterns and local exponential stability of neural networks. The results were used to design efficient continuo...Analytical techniques and Liapunov method were used for the estimation of the attraction domain of memory patterns and local exponential stability of neural networks. The results were used to design efficient continuous feedback associative memory neural networks. The neural network synthesis procedure ensured the gain of large exponential convergence rate without reduction of the attraction domain.展开更多
By using the method of dynamical systems to the two-component generalization of the Camassa-Holm equation, the existence of solitary wave solutions, kink and anti-kink wave solutions, and uncountably infinite many bre...By using the method of dynamical systems to the two-component generalization of the Camassa-Holm equation, the existence of solitary wave solutions, kink and anti-kink wave solutions, and uncountably infinite many breaking wave solutions, smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of travelling wave solutions are listed.展开更多
Employ theory of bifurcations of dynamical systems to a system of coupled nonlin-ear equations,the existence of solitary wave solutions,kink wave solutions,anti-kink wave solutions and periodic wave solutions is obtai...Employ theory of bifurcations of dynamical systems to a system of coupled nonlin-ear equations,the existence of solitary wave solutions,kink wave solutions,anti-kink wave solutions and periodic wave solutions is obtained.Under different parametric conditions,various sufficient conditions to guarantee the existence of the above so-lutions are given.Some exact explicit parametric representations of travelling wave solutions are derived.展开更多
In this paper,the bifurcation method of dynamical systems and numerical approach of differential equations are employed to study CH-γequation.Two new types of bounded waves are found.One of them is called the compact...In this paper,the bifurcation method of dynamical systems and numerical approach of differential equations are employed to study CH-γequation.Two new types of bounded waves are found.One of them is called the compacton.The other is called the generalized kink wave.Their planar graphs are simulated and their implicit expressions are given.The identity of theoretical derivation and numerical simulation is displayed.展开更多
Recently,many authors have studied the following CH-γequation:ut+c0ux+3uux-α2(wxxt+uuxxx+2uxuxx)4-γuxxx=0,whereα2,c0 andγare paramters.Its bounded wave solutions have been investigated mainly for the caseα2>0...Recently,many authors have studied the following CH-γequation:ut+c0ux+3uux-α2(wxxt+uuxxx+2uxuxx)4-γuxxx=0,whereα2,c0 andγare paramters.Its bounded wave solutions have been investigated mainly for the caseα2>0.For the caseα2<0,the existence of three bounded waves(regular solitary waves,compactons,periodic peakons)was pointed out by Dullin et al.But the proof has not been given.In this paper,not only the existence of four types of bounded waves:periodic waves,compacton-like waves,kink-like waves,regular solitary waves,is shown,but also their explicit expressions or implicit expressions are given for the caseα2<0.Some planar graphs of the bounded wave solutions and their numerical simulations are given to show the correctness of our results.展开更多
In this paper, we use phase plane analysis to study the compactons of the nonlinear equation. Four new implicit expressions of the compactons are obtained. These new implicit expressions are given by inverse tangent f...In this paper, we use phase plane analysis to study the compactons of the nonlinear equation. Four new implicit expressions of the compactons are obtained. These new implicit expressions are given by inverse tangent functions. Our work extends previous results. For two sets of the data, the graphs of the implicit functions are drawn and numerical simulations are given to test the correctness of our theoretical results.展开更多
A In this paper, we employ both bifurcation method of dynamical systems and numerical exploration of differential equations to investigate the periodic waves of a general compressible hyperelastic rod equation ut＋3uu...A In this paper, we employ both bifurcation method of dynamical systems and numerical exploration of differential equations to investigate the periodic waves of a general compressible hyperelastic rod equation ut＋3uux-uxxt-γ（2uxuxx＋uuxxx）=0, with parameter γ 〈 0. New expressions including explicit expressions and implicit expressions are obtained. Some previous results are extended. Specially, a new phenomenon is found： when the initial value tends to a certain number, the periodic shock wave suddenly changes into a smooth periodic wave. In dynamical systems, this represents that one of orbits can pass through the singular line. The coherency of numerical simulation and theoretical derivation implies the correctness of our results.展开更多
基金Supported by the National Natural Science Foundation of China under Grant Nos.11171041 and 10971018the Natural Science Foundation of Shandong Province under Grant No.ZR2010AM029+1 种基金the Promotive Research Fund for Young and Middle-Aged Scientists of Shandong Province under Grant No.BS2010SF001 the Doctoral Foundation of Binzhou University under Grant No.2009Y01
文摘This paper is concerned with the generalized nonlinear second-order equation.By the direct construction method,all of the first-order multipliers of the equation are obtained,and the corresponding complete conservation laws(CLs) of such equations are provided.Furthermore,the integrability of the equation is considered in terms of the conservation laws.In addition,the relationship of multipliers and symmetries of the equations is investigated.
文摘By using the theory of bifurcations of planar dynamic systems to the coupled Jaulent-Miodek equations, the existence of smooth solitary travelling wave solutions and uncountably infinite many smooth periodic travelling wave solutions is studied and the bifurcation parametric sets are shown. Under the given parametric conditions, all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained.
文摘By using the theory of the generalized perturbed Hamiltonian systems, it is shown that there exist periodic stream lines in the three_dimensional square cell pattern of Rayleigh_Benard convection. The result means that our method enables this three_dimensional flow pattern to be described in an unambiguous manner, and some experimental results of other authors can be explained.
基金supported by the National Natural Science Foundation of China (No. 10971085)
文摘By using the theory of planar dynamical systems to the ion acoustic plasma equations, we obtain the existence of the solutions of the smooth and non-smooth solitary waves and the uncountably infinite smooth and non-smooth periodic waves. Under the given parametric conditions, we present the sufficient conditions to guarantee the existence of the above solutions.
文摘Analytical techniques and Liapunov method were used for the estimation of the attraction domain of memory patterns and local exponential stability of neural networks. The results were used to design efficient continuous feedback associative memory neural networks. The neural network synthesis procedure ensured the gain of large exponential convergence rate without reduction of the attraction domain.
基金the National Natural Science Foundation of China (10671179) and (10772158)
文摘By using the method of dynamical systems to the two-component generalization of the Camassa-Holm equation, the existence of solitary wave solutions, kink and anti-kink wave solutions, and uncountably infinite many breaking wave solutions, smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of travelling wave solutions are listed.
基金the Natural Science Foundation of Yunnan Province(2006B0081M).
文摘Employ theory of bifurcations of dynamical systems to a system of coupled nonlin-ear equations,the existence of solitary wave solutions,kink wave solutions,anti-kink wave solutions and periodic wave solutions is obtained.Under different parametric conditions,various sufficient conditions to guarantee the existence of the above so-lutions are given.Some exact explicit parametric representations of travelling wave solutions are derived.
基金supported by the National Natural Science Foundation of China(Grant No.10261008).
文摘In this paper,the bifurcation method of dynamical systems and numerical approach of differential equations are employed to study CH-γequation.Two new types of bounded waves are found.One of them is called the compacton.The other is called the generalized kink wave.Their planar graphs are simulated and their implicit expressions are given.The identity of theoretical derivation and numerical simulation is displayed.
基金This work was supported by the National Natural Science Foundation of China(Grant No.10571062)
文摘Recently,many authors have studied the following CH-γequation:ut+c0ux+3uux-α2(wxxt+uuxxx+2uxuxx)4-γuxxx=0,whereα2,c0 andγare paramters.Its bounded wave solutions have been investigated mainly for the caseα2>0.For the caseα2<0,the existence of three bounded waves(regular solitary waves,compactons,periodic peakons)was pointed out by Dullin et al.But the proof has not been given.In this paper,not only the existence of four types of bounded waves:periodic waves,compacton-like waves,kink-like waves,regular solitary waves,is shown,but also their explicit expressions or implicit expressions are given for the caseα2<0.Some planar graphs of the bounded wave solutions and their numerical simulations are given to show the correctness of our results.
基金Supported by the National Natural Science Foundation of China （No.10571062 10371037）.Acknowledgment The first author thanks to the Department of Science and Technology of Yuxi City for its support for doing this work.
文摘In this paper, we use phase plane analysis to study the compactons of the nonlinear equation. Four new implicit expressions of the compactons are obtained. These new implicit expressions are given by inverse tangent functions. Our work extends previous results. For two sets of the data, the graphs of the implicit functions are drawn and numerical simulations are given to test the correctness of our theoretical results.
基金Research is supported by the National Natural Science Foundation of China （No.10571062）.
文摘A In this paper, we employ both bifurcation method of dynamical systems and numerical exploration of differential equations to investigate the periodic waves of a general compressible hyperelastic rod equation ut＋3uux-uxxt-γ（2uxuxx＋uuxxx）=0, with parameter γ 〈 0. New expressions including explicit expressions and implicit expressions are obtained. Some previous results are extended. Specially, a new phenomenon is found： when the initial value tends to a certain number, the periodic shock wave suddenly changes into a smooth periodic wave. In dynamical systems, this represents that one of orbits can pass through the singular line. The coherency of numerical simulation and theoretical derivation implies the correctness of our results.
