  In this paper, we prove the existence of global solutions in
$H^3(\mathbb{R})\cap H^{2,1}(\mathbb{R})$ to the Fokas-Lenells (FL) equation on
the line when the initial data includes solitons.A key tool in proving this
result is a newly modified Darboux transformation, which adds or subtracts a
soliton with given spectral and scattering parameters. In this way the inverse
scattering transform technique is then applied to establish the global
well-posedness of initial value problem with a finite number of solitons based
on our previous results on the global well-posedness of the FL equation.
