  We report a detailed semi-analytical treatment to investigate the dynamics of
a single cavity soliton (CS) and two co-propagating CSs separately in a Kerr
mediated passive optical fiber resonator which is driven by a phase-modulated
pump. The perturbation is dealt with by introducing a Rayleigh's dissipation
function in the framework of variational principle that results in a set of
coupled ordinary differential equations describing the evolution of individual
soliton parameters. We further derive closed-form expressions for quick
estimation of the temporal trajectory, drift velocity and the phase shift
accumulated by the CS due to the externally modulated pump. We also extend the
variational approach to solve two solitons interaction problem in the absence
as well as in the presence of the externally modulated field. In absence of
phase modulated field, the two copropagating solitons can attract, repulse or
can propagate independently depending on their initial delay. The final state
of interaction can be predicted through a second-order differential equation
which is derived by the variational method. While in presence of the phase
modulated field, the two solitons interaction can result in annihilation,
merging, breathing or two soliton state depending on the detuning frequency and
the pump power. Variational treatment analytically predicts these states and
portrays the related dynamics that agrees with full numerical simulation
carried out by solving the normalized Lugiato-Lefever equation. The results
obtained through this variational approach will enrich the understanding of
complex pulse dynamics under phase modulated driving field in passive
dissipative systems.
