  Modifications of the non-linear Schr\"odinger model (MNLS) $ i \partial_{t}
\psi(x,t) + \partial^2_{x} \psi(x,t) - [\frac{\delta V}{\delta |\psi|^2} ]
\psi(x,t) = 0,$ where $\psi \in C$ and $V: R_{+} \rightarrow R$, are
considered. We show that the MNLS models possess infinite towers of
quasi-conservation laws for soliton-type configurations with a special complex
conjugation, shifted parity and delayed time reversion (${\cal C}{\cal
P}_s{\cal T}_d$) symmetry. Infinite towers of anomalous charges appear even in
the standard NLS model for ${\cal C}{\cal P}_s{\cal T}_d$ invariant $N-$bright
solitons. The true conserved charges emerge through some kind of anomaly
cancellation mechanism. A dual Riccati-type pseudo-potential formulation is
introduced for a modified AKNS system (MAKNS) and infinite towers of novel
anomalous conservation laws are uncovered. In addition, infinite towers of
exact non-local conservation laws are uncovered in a linear system formulation.
Our analytical results are supported by numerical simulations of
$2-$bright-soliton scatterings with potential $ V = - \frac{ 2\eta}{2+
\epsilon} ( |\psi|^2 )^{2 + \epsilon}, \epsilon \in R, \eta>0$. Our numerical
simulations show the elastic scattering of bright solitons for a wide range of
values of the set $\{\eta, \epsilon\}$ and a variety of amplitudes and relative
velocities. The AKNS-type system is quite ubiquitous, and so, our results may
find potential applications in several areas of non-linear physics, such as
Bose-Einstein condensation, superconductivity, soliton turbulence and the
triality among gauge theories, integrable models and gravity theories.
