Perturbation theory for solitons of the Fokas--Lenells equation : Inverse scattering transform approach

We present perturbation theory based on the inverse scattering transform method for solitons described by an equation with the inverse linear dispersion law $\omega \sim 1\mathrm{/}k$, 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.