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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 $V: R_{+} \rightarrow R$, areconsidered. We show that the MNLS models possess infinite towers ofquasi-conservation laws for soliton-type configurations with a special complexconjugation, shifted parity and delayed time reversion (${\cal C}{\calP}_s{\cal T}_d$) symmetry. Infinite towers of anomalous charges appear even inthe standard NLS model for ${\cal C}{\cal P}_s{\cal T}_d$ invariant $N-$brightsolitons. The true conserved charges emerge through some kind of anomalycancellation mechanism. A dual Riccati-type pseudo-potential formulation isintroduced for a modified AKNS system (MAKNS) and infinite towers of novelanomalous conservation laws are uncovered. In addition, infinite towers ofexact 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 numericalsimulations show the elastic scattering of bright solitons for a wide range ofvalues of the set $\{\eta, \epsilon\}$ and a variety of amplitudes and relativevelocities. The AKNS-type system is quite ubiquitous, and so, our results mayfind potential applications in several areas of non-linear physics, such asBose-Einstein condensation, superconductivity, soliton turbulence and thetriality among gauge theories, integrable models and gravity theories.

Modified non-linear Schr\"odinger models, ${\cal C}{\cal P}_s{\cal T}_d$ invariant $N-$bright solitons and infinite towers of anomalous charges