  The channel law for amplitude-modulated solitons transmitted through a
nonlinear optical fibre with ideal distributed amplification and a receiver
based on the nonlinear Fourier transform is a noncentral chi-distribution with
$2n$ degrees of freedom, where $n=2$ and $n=3$ correspond to the single- and
dual-polarisation cases, respectively. In this paper, we study capacity lower
bounds of this channel under an average power constraint in bits per channel
use. We develop an asymptotic semi-analytic approximation for a capacity lower
bound for arbitrary $n$ and a Rayleigh input distribution. It is shown that
this lower bound grows logarithmically with signal-to-noise ratio (SNR),
independently of the value of $n$. Numerical results for other continuous input
distributions are also provided. A half-Gaussian input distribution is shown to
give larger rates than a Rayleigh input distribution for $n=1,2,3$. At an SNR
of $25$ dB, the best lower bounds we developed are approximately $3.68$ bit per
channel use. The practically relevant case of amplitude shift-keying (ASK)
constellations is also numerically analysed. For the same SNR of $25$ dB, a
$16$-ASK constellation yields a rate of approximately $3.45$ bit per channel
use.
