This article studies bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solution...This article studies bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solutions for the above equation by the theory and method of planar dynamical systems, and obtain their existent conditions, number, and general shape. Secondly, we investigate the dissipation effect on the shape evolution of bounded traveling wave solutions. We find out a critical value r^＊ which can characterize the scale of dissipation effect, and prove that the bounded traveling wave solutions appear as kink profile waves if ｜r｜≥ r^＊; while they appear as damped oscillatory waves if ｜r｜ 〈 r^＊. We also obtain kink profile solitary wave solutions with and without dissipation effect. On the basis of the above discussion, we sensibly design the structure of the approximate damped oscillatory solutions according to the orbits evolution relation corresponding to the component u（ξ） in the global phase portraits, and then obtain the approximate solutions （u（ξ）, H（ξ））. Furthermore, by using homogenization principle, we give their error estimates by establishing the integral equation which reflects the relation between exact and approximate solutions. Finally, we discuss the dissipation effect on the amplitude, frequency, and energy decay of the bounded traveling wave solutions.展开更多
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
基金supported by National Natural ScienceFoundation of China(11071164)Innovation Program of Shanghai Municipal Education Commission(13ZZ118)Shanghai Leading Academic Discipline Project(XTKX2012)
文摘This article studies bounded traveling wave solutions of variant Boussinesq equation with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solutions for the above equation by the theory and method of planar dynamical systems, and obtain their existent conditions, number, and general shape. Secondly, we investigate the dissipation effect on the shape evolution of bounded traveling wave solutions. We find out a critical value r^＊ which can characterize the scale of dissipation effect, and prove that the bounded traveling wave solutions appear as kink profile waves if ｜r｜≥ r^＊; while they appear as damped oscillatory waves if ｜r｜ 〈 r^＊. We also obtain kink profile solitary wave solutions with and without dissipation effect. On the basis of the above discussion, we sensibly design the structure of the approximate damped oscillatory solutions according to the orbits evolution relation corresponding to the component u（ξ） in the global phase portraits, and then obtain the approximate solutions （u（ξ）, H（ξ））. Furthermore, by using homogenization principle, we give their error estimates by establishing the integral equation which reflects the relation between exact and approximate solutions. Finally, we discuss the dissipation effect on the amplitude, frequency, and energy decay of the bounded traveling wave solutions.
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.
