  We present perturbation theory based on the inverse scattering transform
method for solitons described by an equation with the inverse linear dispersion
law $\omega\sim 1/k$, where $\omega$ is the frequency and $k$ is the wave
number, and cubic nonlinearity. This equation, first suggested by Davydova and
Lashkin for describing dynamics of nonlinear short-wavelength ion-cyclotron
waves in plasmas and later known as the Fokas--Lenells equation, arises from
the first negative flow of the Kaup--Newell hierarchy. Local and nonlocal
integrals of motion, in particular the energy and momentum of nonlinear
ion-cyclotron waves, are explicitly expressed in terms of the discrete
(solitonic) and continuous (radiative) scattering data. Evolution equations for
the scattering data in the presence of a perturbation are presented. Spectral
distributions in the wave number domain of the energy emitted by the soliton in
the presence of a perturbation are calculated analytically for two cases: (i)
linear damping that corresponds to Landau damping of plasma waves, and (ii)
multiplicative noise which corresponds to thermodynamic fluctuations of the
external magnetic field (thermal noise) and/or the presence of a weak plasma
turbulence.
