density.

In the absence of perturbation, $P(v_i,v_j)=0$, the solution Eq. (\[soliton\]) describes overlapping solitons moving at constant velocity, and hence $\ddot z_0=\dot\varphi_i=0$. However, the effect of the perturbation $P(v_i,v_j)$ leads to the adiabatic time evolution of the soliton parameters, $(\varphi, z_0)\rightarrow (\varphi(t), z_0(t))$. This evolution can be described by the equation [@Kivshar1994] $$\begin{aligned}
\dot\varphi_i=\frac{g_{12}/g}{2 \cos^2 \varphi_i \sin \varphi_i}
\Re\left[\int_{-\infty}^{\infty} P(v_i,v_j)\frac{\partial v_i^*}{\partial t} 
dz\right].
\label{pert}\end{aligned}$$

Assuming solitons with high density depletions $\varphi_i<<1$ and separated by a relative distance $2z_0$, such that $\zeta_1=\cos \varphi_1 
\left(z-z_0\right)$ and $\zeta_2=\cos \varphi_2 \left(z+z_0\right)$, Eq. (\[pert\]) gives $$\begin{aligned}
\dot\varphi_1=-\frac{g_{12}}{g}f(z_0)\equiv \, F_{\rm eff}^{(i)}(z_0),
\label{eq:phi}\end{aligned}$$ where $f(z_0)=8\sinh(2z_0)e^{6z_0}(3-3e^{8z_0}+(1+4e^{4z_0}+e^{8z_0})4z_0)/(e^{4z_0}
-1)^5$. As can be seen, for $z_0=0$, that is for overlapped solitons, $\dot \varphi_1=0$ and the solitons evolve without relative motion. Since, in this limit ($\varphi<<1$) $\dot z_0\approx 
\varphi$, Eq. (\[eq:phi\]) is also an equation for $\ddot z_0$, with the right hand side playing the role of a classical force $F_{\rm eff}^{(i)}(z_0)$ derived from an effective potential $U_{\rm eff}^{(i)}(z_0)$ : $$\begin{aligned}
\ddot z_0=-\frac{d U_{\rm eff}^{(i)}(z_0)}{dz_0}.
\label{eq:rel}\end{aligned}$$ This potential accounts for the interactions between the solitons and is the source of their relative motion. After integration of the force Eq. (\[eq:phi\]) over $z_0$, we get $$\begin{aligned}
U_{\rm eff}^{(i)}(z_0)=\frac{g_{12}}{g}\,\frac{2(1+e^{4z_0})(1-2z_0)-4}
{ e^{-4z_0}\,(e^{4z_0}-1)^3}.
\label{eq:int_pot}\end{aligned}$$ As it 