  Under investigation is the nonlinear Schr\"odinger equation hierarchies and
the reversible transformations. We propose a generalized reversible
transformation between the the generalized NLSE hierarchy with focussing and
defocussing nonlinearity and the NLSE hierarchy forced with a linear potential
term. The corresponding extended concept of classical dark and bright solitons
of the forced hierarchy, accelerating due to linear potential as well as due to
the dispersion are obtained directly without resolving the nonisospectral
inverse scattering problem. We have identified a set of new constraints among
the dispersion and the nonlinear coefficients in the inhomogeneous NLSE
hierarchy, which are preserved after the transformations. The reversible
transformations allow us to encompass inhomogeneous NLS, HNLS and higher order
equations belonging to the class of nonisospectral family of inverse scattering
problems to the isospectral NLS class of equations and study them under a
general mathematical framework. We hope that our analysis provides a
mathematical platform to study inhomogeneous NLSEs as well as open up the
possibility of new applications in physics.
