is linear in $p_X(\lambda )$ and positive, the time-scaled ${\mathsf{I}(X;Y)}$ is quasiconcave [@stancu-minasian_frac_programming Table 2.5.2]. We can solve [()]{} and obtain the corresponding capacity-achieving distribution numerically.

![Optimal distribution for different .[]{data-label="fig:optDist:example_opt_dist"}](optimal_distribution_combined_sufigures "fig:") -0.5cm

Exemplary results of the capacity-achieving distribution are shown in [Fig.]{}. We note that the lowest and highest amplitudes are always used with equal and high probability. For low , only these are used, i.e., is optimal. Furthermore, the capacity-achieving distribution is discrete and is of exponential-like shape with the exception of a point mass at zero as it can be seen in [Fig.]{}.

Note that ${\mathsf{C}}$ assumes memorylessness, which does not necessarily hold due to the variable symbol interval. Hence, ${\mathsf{C}}$ is, in fact, the constraint capacity under the assumption of a memoryless channel and the constraint of transmitting only first-order solitons. However, for notational simplicity, we refer to it simply as capacity with its corresponding capacity-achieving distribution.

In the case of a noiseless channel, it is possible to derive a closed form solution to [()]{} under the assumption of a finite discretization.

\[lemma:noiseless\] Let $\lambda_1, \lambda_2, \ldots, \lambda_M$ be $M \geq 2$ eigenvalues with $0 \leq {\mathfrak{I}\{\lambda_1\}} < {\mathfrak{I}\{\lambda_2\}} < \ldots < {\mathfrak{I}\{\lambda_M\}}$ and let ${T{\mathopen{}\mathclose\bgroup\originalleft}(\lambda_k{\aftergroup\egroup\originalright})}$ 