  We consider modifications of the nonlinear Schrodinger model (NLS) to look at
the recently introduced concept of quasi-integrability. We show that such
models possess an infinite number of quasi-conserved charges which present
intriguing properties in relation to very specific space-time parity
transformations. For the case of two-soliton solutions where the fields are
eigenstates of this parity, those charges are asymptotically conserved in the
scattering process of the solitons. Even though the charges vary in time their
values in the far past and the far future are the same. Such results are
obtained through analytical and numerical methods, and employ adaptations of
algebraic techniques used in integrable field theories. Our findings may have
important consequences on the applications of these models in several areas of
non-linear science. We make a detailed numerical study of the modified NLS
potential of the form V = |psi|^(2(2+epsilon)), with epsilon being a
perturbation parameter. We perform numerical simulations of the scattering of
solitons for this model and find a good agreement with the results predicted by
the analytical considerations. Our paper shows that the quasi-integrability
concepts recently proposed in the context of modifications of the sine-Gordon
model remain valid for perturbations of the NLS model.
