  The global existence of the solution for the second-type derivative nonlinear
Schr\"odinger (DNLSII) equation with solitons is presented for the first time
on the line with weighted Sobolev initial data in $H^2( \mathbb{R}) \cap
H^{1,1}(\mathbb{R} )$.
  The modified Darboux transformation is applied as a main research tool to
analyze the global existence for the Cauchy problem by adding or subtracting a
finite number of zeros of the scattering data. For this purpose, it is
necessary to study the invertibility and uniqueness of the Darboux
transformation first acting on the weighted Sobolev space.
  The analysis of time evolution comprehensively demonstrates the existence of
a unique global solution to the DNLSII equation with Cauchy problem in $H^2(
\mathbb{R}) \cap H^{1,1}(\mathbb{R} )$.
