  Droplet solitons are coherently precessing solitary waves that have been
recently realized in thin ferromagnets with perpendicular anisotropy.In the
strongly nonlinear regime, droplets can be well approximated by a slowly
precessing, circular domain wall with a hyperbolic tangent form. Utilizing this
representation, this work develops a general droplet modulation theory and
applies it to study the long range effects of the magnetostatic field and a
nanocontact spin torque oscillator (NC-STO) where spin polarized currents act
as a gain source to counteract magnetic damping. An analysis of the dynamical
equations for the droplet's center, frequency and phase demonstrates a negative
processional frequency shift due to long range dipolar interactions, dependent
on film thickness. Further analysis also demonstrates the onset of a
saddle-node bifurcation at the minimum sustaining current for the NC-STO. The
basin of attraction associated with the stable node demonstrates that spin
torque enacts a restoring force to excursions of the droplet from the
nanocontact center, observed previously in numerical simulations. Large
excursions lead to the droplet's eventual decay into spin waves.
