论文部分内容阅读
本文是一篇综述.首先,我们用动力系统方法和奇行波方程理论研究熟知的广义Camassa-Holm方程与Degasperis-Procesi方程的精确行波解,给出尖孤子解,伪尖孤子解,周期尖波,伪周期尖波和紧支集解族的参数表示.这些精确解说明,在给定的参数条件下,当系统的能量改变时,尖孤子解是一族伪周期尖波的极限解;而当参数改变时,尖孤子解是一族周期尖波或一族伪尖孤子波的极限解.伪周期尖波与伪尖孤子波是有两时间尺度的光滑经典解.第二,我们用几类非线性波方程模型的精确解说明,不同于广义Camassa-Holm方程与Degasperis-Procesi方程的精确尖孤子解,存在各种不同形式的精确的尖孤子的参数表示.第三,针对近年国际数学刊物发表的有些作者的“peakon equations(尖孤子方程)”的文章,我们指出,对应于所谓尖孤子方程,其行波方程是第二类奇行波方程,这类方程没有尖孤子解.“,”In this paper,we first study the exact peakon,periodic peakon,pseudo-peakon as well as the compacton solutions for the generalized Camassa-Holm equation and the Degasperis-Procesi equation.Based on the method of dynamical systems and the theory of singular traveling wave equations,the exact explicit parametric representations of the above mentioned solutions are derived.These solutions tell us that peakon is a limit solution of a family of periodic peakons or a limit solution of a family of pseudo-peakons under two classes of limit senses.The pseudo-peakon and pseudo-periodic peakon family are smooth classical solutions with two time scales.Second,we use some nonlinear wave equation models to show that there exist various exact explicit peakon solutions,which are different from the peakon solutions given by the generalized Camassa-Holm equation and the Degasperis-Procesi equation.Third,we point out that the so called“peakon equations”in some references have no peakons.Corresponding to these“peakon equations”,their traveling systems are the singular traveling systems of the second kind,which can not have the peakon solution.