  The capacity of the channel defined by the stochastic nonlinear Schr\"odinger
equation, which includes the effects of the Kerr nonlinearity and amplified
spontaneous emission noise, is considered in the case of zero dispersion. In
the absence of dispersion, this channel behaves as a collection of parallel
per-sample channels. The conditional probability density function of the
nonlinear per-sample channels is derived using both a sum-product and a
Fokker-Planck differential equation approach. It is shown that, for a fixed
noise power, the per-sample capacity grows unboundedly with input signal. The
channel can be partitioned into amplitude and phase subchannels, and it is
shown that the contribution to the total capacity of the phase channel declines
for large input powers. It is found that a two-dimensional distribution with a
half-Gaussian profile on the amplitude and uniform phase provides a lower bound
for the zero-dispersion optical fiber channel, which is simple and
asymptotically capacity-achieving at high signal-to-noise ratios (SNRs). A
lower bound on the capacity is also derived in the medium-SNR region. The exact
capacity subject to peak and average power constraints is numerically
quantified using dense multiple ring modulation formats. The differential model
underlying the zero-dispersion channel is reduced to an algebraic model, which
is more tractable for digital communication studies, and in particular it
provides a relation between the zero-dispersion optical channel and a $2 \times
2$ multiple-input multiple-output Rician fading channel. It appears that the
structure of the capacity-achieving input distribution resembles that of the
Rician fading channel, i.e., it is discrete in amplitude with a finite number
of mass points, while continuous and uniform in phase.
