Resonant transmission of a soliton across an interface between two Toda lattices

The transmission of a single soliton is investigated numerically across an interface between two Toda lattices which are connected by a harmonic spring. We find that a resonant transmission of the soliton occurs when the spring constant of the harmonic spring is adjusted properly. Furthermore, when the amplitude of the incident soliton is large, the soliton transmission coefficient exhibits a local minimum which is due to an emergence of localized waves around the harmonic spring. We propose an experimental test of the results by using a nonlinear $LC$ circuit.

Figure 1

The perspective of an incident soliton on two Toda lattices connected by a harmonic spring.

Figure 2

The spatiotemporal evolution of energy density $h_n\left(\tau \right)$ which is normalized by total energy. $\tau$ is dimensionless time. The wave number $k_0$ of the incident soliton is $k_0=2.5$. The spring constants $\kappa$ of the harmonic spring are (a) $\kappa =38.54$ and (b) $\kappa =7.776$.

Figure 3

The frontier soliton transmission coefficient $T_1$ as a function of the spring constant $\kappa$ of the harmonic spring in the logarithmic scale. The wave numbers of the incident soliton are $k_0=1(—)$, 1.5(---), 2(⋯) and 2.5(- ∙ ‐).

Figure 4

The spring constants ${\kappa }_{\max }$ of the harmonic spring at the maximum of $T_1$ (●) are plotted as a function of the wave number $k_0$ of the incident soliton. The solid curve is the fit by Eq. (15). The broken curve is the approximant given by Eq. (18).

Figure 5

The temporal localized energy $L\left(\tau \right)$ is plotted as a function of dimensionless time $\tau$ for $\kappa =3(—)$ and 30(---). The wave number of the incident soliton is $k_0=2.5$.

Figure 6

The frontier soliton transmission coefficient $T_1(—)$, the localized energy $L\left(\text{‐}\text{‐}\text{‐}\right)$, the reflected energy $R(\cdots )$ and the energy of the ripples $T_{\mathrm{ri}}(\text{‐}\bullet \text{‐})$ are plotted against $\kappa$. The wave number of the incident soliton is $k_0=2.5$.