  The Heimburg-Jackson model, or thermodynamic soliton theory of nervous
impulses, has a well-established record as an alternative model for studying
the dynamics of nerve impulses and lipid bilayers. Within this framework, nerve
impulses can be represented as nonlinear excitations of low amplitude depicted
by the damped nonlinear Schr\"odinger equation and their adiabatic evolution
can be analyzed using direct perturbative methods. Based on the foregoing, we
carry out the current study using the quasi-stationary approach to obtain the
adiabatic evolution of solitons embedded in lipid bilayers under the influence
of a viscous elastic fluid. This analysis encompasses liquid-to-gel transition
of the lipid bilayers, for whose dark and bright solitons arise, respectively.
