Modified nonlinear Schrödinger models,

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