This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of pla...This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.展开更多
This paper deals with the problem of the bounded traveling wave solutions' shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equation for shor...This paper deals with the problem of the bounded traveling wave solutions' shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equation for short.The authors employ the theory and method of planar dynamical systems to make comprehensive qualitative analyses to the above equation satisfied by the horizontal velocity component u(ξ) in the traveling wave solution (u(ξ),H(ξ)),and then give its global phase portraits.The authors obtain the existent conditions and the number of the solutions by using the relations between the components u(ξ) and H(ξ) in the solutions.The authors study the dissipation effect on the solutions,find out a critical value r*,and prove that the traveling wave solution (u(ξ),H(ξ)) appears as a kink profile solitary wave if the dissipation effect is greater,i.e.,|r| ≥ r*,while it appears as a damped oscillatory wave if the dissipation effect is smaller,i.e.,|r| < r*.Two solitary wave solutions to the WBK equation without dissipation effect is also obtained.Based on the above discussion and according to the evolution relations of orbits corresponding to the component u(ξ) in the global phase portraits,the authors obtain all approximate damped oscillatory solutions (u(ξ),H(ξ)) under various conditions by using the undetermined coefficients method.Finally,the error between the approximate damped oscillatory solution and the exact solution is an infinitesimal decreasing exponentially.展开更多
In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter cond...In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter conditions. Then, the relations between the behaviors of bounded traveling wave solutions and the dissipation coeffiicient μ are investigated. We find that a bounded traveling wave solution appears as a kink profile solitary wave solution when μ is more than the critical value obtained in this paper, while a bounded traveling wave solution appears as a damped oscillatory solution when μ is less than it. Furthermore, we explain the solitary wave solutions obtained in previous literature, and point out their positions in global phase portraits. In the meantime, approximate damped oscillatory solutions are given by means of undetermined coefficients method. Finally, based on integral equations that reflect the relations between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate solutions are presented.展开更多
In this paper,we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical syste...In this paper,we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems.Two critical values which can characterize the scale of dissipation effect are obtained.If dissipation effect is not less than a certain critical value,the traveling wave solutions appear as kink profile;while if it is less than this critical value,they appear as damped oscillatory.All expressions of bounded traveling wave solutions are presented,including exact expressions of bell and kink profile solitary wave solutions,as well as approximate expressions of damped oscillatory solutions.For approximate damped oscillatory solution,using homogenization principle,we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions.It can be seen that the error is an infinitesimal decreasing in the exponential form.展开更多
基金Project supported by the National Natural Science Foundation of China(No.11071164)the Innovation Program of Shanghai Municipal Education Commission(No.13ZZ118)+1 种基金the Shanghai Leading Academic Discipline Project(No.XTKX2012)the Innovation Fund Project for Graduate Stu-dent of Shanghai(No.JWCXSL1201)
文摘This paper aims at analyzing the shapes of the bounded traveling wave solu- tions for a class of nonlinear wave equation with a quintic term and obtaining its damped oscillatory solutions. The theory and method of planar dynamical systems are used to make a qualitative analysis to the planar dynamical system which the bounded traveling wave solutions of this equation correspond to. The shapes, existent number, and condi- tions are presented for all bounded traveling wave solutions. The bounded traveling wave solutions are obtained by the undetermined coefficients method according to their shapes, including exact expressions of bell and kink profile solitary wave solutions and approxi- mate expressions of damped oscillatory solutions. For the approximate damped oscillatory solution, using the homogenization principle, its error estimate is given by establishing the integral equation, which reflects the relation between the exact and approximate so- lutions. It can be seen that the error is infinitesimal decreasing in the exponential form.
基金Project supported by the National Natural Science Foundation of China (No.11071164)the Natural Science Foundation of Shanghai (No.10ZR1420800)the Shanghai Leading Academic Discipline Project (No.S30501)
文摘This paper deals with the problem of the bounded traveling wave solutions' shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equation for short.The authors employ the theory and method of planar dynamical systems to make comprehensive qualitative analyses to the above equation satisfied by the horizontal velocity component u(ξ) in the traveling wave solution (u(ξ),H(ξ)),and then give its global phase portraits.The authors obtain the existent conditions and the number of the solutions by using the relations between the components u(ξ) and H(ξ) in the solutions.The authors study the dissipation effect on the solutions,find out a critical value r*,and prove that the traveling wave solution (u(ξ),H(ξ)) appears as a kink profile solitary wave if the dissipation effect is greater,i.e.,|r| ≥ r*,while it appears as a damped oscillatory wave if the dissipation effect is smaller,i.e.,|r| < r*.Two solitary wave solutions to the WBK equation without dissipation effect is also obtained.Based on the above discussion and according to the evolution relations of orbits corresponding to the component u(ξ) in the global phase portraits,the authors obtain all approximate damped oscillatory solutions (u(ξ),H(ξ)) under various conditions by using the undetermined coefficients method.Finally,the error between the approximate damped oscillatory solution and the exact solution is an infinitesimal decreasing exponentially.
基金supported by Shanghai Leading Academic Discipline Project (No. S30501)Shanghai Natural Science Foundation Project (No. 10ZR1420800)
文摘In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter conditions. Then, the relations between the behaviors of bounded traveling wave solutions and the dissipation coeffiicient μ are investigated. We find that a bounded traveling wave solution appears as a kink profile solitary wave solution when μ is more than the critical value obtained in this paper, while a bounded traveling wave solution appears as a damped oscillatory solution when μ is less than it. Furthermore, we explain the solitary wave solutions obtained in previous literature, and point out their positions in global phase portraits. In the meantime, approximate damped oscillatory solutions are given by means of undetermined coefficients method. Finally, based on integral equations that reflect the relations between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate solutions are presented.
基金This research is supported by the National Natural Science Foundation of China(No.11071164)Shanghai Natural Science Foundation Project(No.10ZR1420800)Leading Academic Discipline Project of Shanghai Municipal Government(No.S30501).
文摘In this paper,we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems.Two critical values which can characterize the scale of dissipation effect are obtained.If dissipation effect is not less than a certain critical value,the traveling wave solutions appear as kink profile;while if it is less than this critical value,they appear as damped oscillatory.All expressions of bounded traveling wave solutions are presented,including exact expressions of bell and kink profile solitary wave solutions,as well as approximate expressions of damped oscillatory solutions.For approximate damped oscillatory solution,using homogenization principle,we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions.It can be seen that the error is an infinitesimal decreasing in the exponential form.
