pulse over the smallest support containing a fraction $(1-\delta ) $ of the energy of the pulse and hence, we can formally define the pulse width as follows.

The pulse width of $\lambda$ is defined as the smallest support containing a fraction $(1-\delta ) $ of the energy of the pulse, $$\begin{aligned}
    T(\lambda, \delta ) &\triangleq \frac{1}{2{\mathfrak{I}\{\lambda\}}} \ln{\mathopen{}\mathclose\bgroup\originalleft}(\frac{2}{\delta} -1{\aftergroup\egroup\originalright}),\end{aligned}$$ where $0 < \delta < 1$.

The value of the cutoff parameter $\delta$ must be chosen in a way such that soliton-soliton interactions are negligible. For longer transmission distances, $\delta$ decreases, i.e., the pulses must be spaced further apart. Furthermore, the condition $$\begin{aligned}
    {\operatorname{e}^{-2{\mathfrak{I}\{\lambda\}} \Delta(\lambda, \delta )}} 
    &= {\operatorname{e}^{-\ln{\mathopen{}\mathclose\bgroup\originalleft}(\frac{2}{\delta} -1{\aftergroup\egroup\originalright})}} \ll 1
    \label{eqn:guard_time_condition}\end{aligned}$$ must be fulfilled [@shevchenko2018tcom].

At this point, it is important to comment on the memorylessness of the system emanating from the absence of soliton-soliton interactions. A pulse train of well-separated first order solitons was investigated in [@shevchenko2018tcom] for launch powers of $\SI{-1.5}{\dBm}$ and $\SI{1.45}{\dBm}$ and transmission over $\SI{500}{\kilo\meter}$ and $\SI{2000}{\kilo\meter}$. It was shown via simulations that the correlation between the symbols at the receiver is essentially zero, concluding that the channel is indeed memoryless in the transmission range of $\SI{500}{\kilo\meter}$ to $\SI{2000}{\kilo\meter}$ and transmit power range of $\SI{-1.5}{\dBm}$ 