  In this work, we study the propagation of solitons in lossy optical fibers.
The main objective of this work is to study the loss of energy of the soliton
wave during propagation and then to evaluate the impact of this loss on the
transmission of the soliton signal. In this context, a numerical scheme was
developed to solve a system of complex partial differential equations (CPDE)
that describes the propagation of solitons in optical fibers with loss and
nonlinear amplification mechanisms. The numerical procedure is based on the
mathematical theory of Taylor series of complex functions. We adapted the
Finite Difference Method (FDM) to approximate derivatives of complex functions.
Then, we solve the algebraic system resulting from the discretization,
implicitly, through the relaxation Gauss-Seidel method (RGSM). The numerical
study of CPDE system with linear and cubic attenuation showed that soliton
waves undergo attenuation, dispersion, and oscillation effects. On the other
hand, we find that by considering the nonlinear term (cubic term) as an optical
amplification, it is possible to partially compensate for the attenuation of
the optical signal. Finally, we show that a gain of 9% triples the propagation
distance of the fundamental soliton wave, when the dissipation rate is 1%.
