  Signal propagation in an optical fiber can be described by the nonlinear
Schr\"odinger equation (NLSE). The NLSE has no known closed-form solution,
mostly due to the interaction of dispersion and nonlinearities. In this paper,
we present a novel closed-form approximate model for the nonlinear optical
channel, with applications to passive optical networks. The proposed model is
derived using logarithmic perturbation in the frequency domain on the
group-velocity dispersion (GVD) parameter of the NLSE. The model can be seen as
an improvement of the recently proposed regular perturbation (RP) on the GVD
parameter. RP and logarithmic perturbation (LP) on the nonlinear coefficient
have already been studied in the literature, and are hereby compared with RP on
the GVD parameter and the proposed LP model. As an application of the model, we
focus on passive optical networks. For a 20 km PON at 10 Gbaud, the proposed
model improves upon LP on the nonlinear coefficient by 1.5 dB. For the same
system, a detector based on the proposed LP model reduces the uncoded
bit-error-rate by up to 5.4 times at the same input power or reduces the input
power by 0.4 dB at the same information rate.

Signal Processing