中文摘要：

研究了一类Toda连续晶格系统的特殊孤立波解：紧孤立波解Compacton和尖峰孤立波解Peakon．设Toda系统中横向与纵向波动处于同一量级，通过行波约化，将Toda系统约化为关于行波变量的常微分方程．假设该方程的解具有局部正弦、局部余弦和指数形式，将常微分方程的求解问题转化为代数方程的求解，利用吴消元法，借助Mathematica数学软件，获得了Toda系统的Compacton解和Peakon解．Compacton解在有限区间外恒为零，是更强局部性的孤立波解．Peakon解在波峰处一阶导数不连续，但可用Dirac广义函数表示．通过电一力类比可以建立与Toda系统等价的电路，利用电路产生的孤子信号可以进行一些特殊的信号处理．

英文摘要：

Special solitary wave solutions of a continuum Toda system are studied, including Compacton solutions and Peakon solutions. By assuming the transversal and longitudinal strains are of the same order of magnitude and introducing a traveling wave parameter, the Toda system is turned to an ordinary differential equation system with respect to the traveling wave parameter. By assuming the solutions are sinusoidal, cosinusoidal and exponential, the problem of solving the ordinary differential system is converted to that of an algebra system. With the aid of Mathematica and Wu elimination method, some special explicit solitary wave solutions,including Compacton solutions and Peakon solutions, are obtained. The solutions are zero outside a finite domain of space variable having a compact support. Peakon solutions have discontinuous first derivative on the peaks and can be described by Dirac function. All these solutions may be applied to some specific tasks of signal processing and communication.