色散项系数为负的MKdV-Burgers方程的有界行波解

利用平面动力系统理论、假设待定法和齐次化原理研究了色散项系数为负的MKdV-Burgers方程的有界行波解,得到了方程行波解所对应的平面动力系统在不同参数条件下的全局相图以及有界行波解存在的条件和个数.讨论了该方程有界行波解的波形与耗散系数之间的关系,给出了表征耗散作用大小的临界值,该临界值与Bikbaev在文献中提出的临界值是不相同的.求出了该方程的钟状和扭状孤波解,进一步根据衰减振荡解对应的解轨线在相图中的演化关系,求得了该方程的衰减振荡解的近似解.给出了所求衰减振荡近似解与精确解的误差估计,其误差是以指数形式速降的无穷小量.最后,证明了所求衰减振荡解的近似解关于对接点的稳定性.

Bounded Traveling Wave Solutions for MKdV-Burgers Equation with Negative Dispersion Coefficient

The bounded traveling wave solutions of MKdV-Burgers equation with negative dispersion coefficient were studied with resorting to the theory of planar dynamical systems,undetermined coefficients method and homogenization principle.The global phase portraits under different parameter conditions as well as the existent number and existense conditions were obtained for the dynamical system corresponding to the traveling wave solutions of the MKdV-Burgers equation.The relation between the profiles of bounded traveling wave solutions and dissipation coefficient was investigated,and a critical value,different from the one proposed by Bikbaev in his article was provided,which characterizes the scale of dissipation effect is presented.The bell and kink profile solitary wave solutions are presented for the MKdV-Burgers equation.Moreover,approximate damped oscillatory solutions of the MKdV-Burgers equation are obtained according to the evolution relation of orbits corresponding to the approximate damped oscillatory solutions in the global phase portraits.This paper also presents the error estimates between exact and approximate solutions for the approximate damped oscillatory solutions.The error is infinitesimal decreasing in exponential form.Finally,the stability at connective points was proved for the approximate damped oscillatory solutions.