A time-dependent generalized non-linear Schrödinger equation (GNLSE) of motion was earlier derived in our laboratory by combining density functional theory and quantum fluid dynamics in threedimensional space. In continuation of the work reported previously, the GNLSE is applied to provide additional knowledge on the femtosecond dynamics of the electron density in the hydrogen molecule interacting with high-intensity laser fields. For this purpose, the GNLSE is solved numerically for many time-steps over a total interaction time of 100 fs, by employing a finite-difference scheme. Various time-dependent (TD) quantities, namely, electron density, ground-state survival probability and dipole moment have been obtained for two laser wavelengths and four different intensities. The high-order harmonics generation (HHG) is also examined. The present approach goes beyond the linear response formalism and, in principle, calculates the TD electron density to all orders of change.