The channel law for amplitude-modulated solitons transmitted through a nonlinear optical fiber 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 the 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 halfGaussian 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 shiftkeying (ASK) constellations is also numerically analyzed. For the same SNR of 25 dB, a 16-ASK constellation yields a rate of approximately 3.45 bit per channel use.