Propagation of solitons in the Toda lattice with an impure segment

The transmission and scattering of a single soliton is studied numerically in the Toda lattice with an impure segment which consists of two kinds of masses. The incident soliton is split into transmitted, reflected, and trapped solitons by the impure segment. The energy of the soliton trapped in the segment escapes from the segment very slowly and thus we can define the transmission rate by the ratio of energies of the transmitted soliton and the incident soliton. It is shown that the dependence of the transmission rate on the segment length N can be fitted quite well by $1/(1+\alpha N^{\beta }).\mathrm{}$ The transmission rate is also shown to be a monotone decreasing function of the wave number of the incident soliton. Most of the energy of the transmitted wave is carried by a large soliton (the frontier soliton) at the front, which is shown to be an exact soliton of the Toda lattice. When the mass difference is small, the transmission rate can be obtained by considering the segment as a repetition of a unit and repeating the renormalization of the wave number due to the unit.