data for the -based transmission system and transmit each pulse as soon as the previous one has been transmitted rather than with a fixed interval as in [@shevchenko2018tcom], yielding increased data rate. Accordingly, we find the capacity-achieving input distribution, maximizing the time-scaled . For ease of notation, we refer to the maximized as capacity noting that it is in fact the constrained capacity of a system transmitting first-order solitons. The scheme then shapes the information symbols according to the capacity-achieving distribution by a . The information symbols are also encoded by a encoder and the parity symbols at the output of the encoder are suitably modulated. The resulting sequence of modulated symbols and the sequence at the output of the are transmitted via time-sharing. We further derive an achievable rate for such a scheme. We demonstrate via discrete-time Monte-Carlo and simulations, that performs at around $\SI{2}{\decibel}$ from capacity using off-the-shelf codes. The proposed scheme yields a significant improvement of up to twice the data rate compared to an unshaped system as in [@shevchenko2018tcom].

It is important to note that although first-order solitons do not outperform conventional coherent systems due to their spectrally inefficient pulse shape compared to a Nyquist pulse 