  Deformations of the focusing non-linear Schr\"odinger model (NLS) are
considered in the context of the quasi-integrability concept. We strengthen the
results of JHEP09(2012)103 for bright soliton collisions. We addressed the
focusing NLS as a complement to the one in JHEP03(2016)005, in which the
modified defocusing NLS models with dark solitons were shown to exhibit an
infinite tower of exactly conserved charges. We show, by means of analytical
and numerical methods, that for certain two-bright-soliton solutions, in which
the modulus and phase of the complex modified NLS field exhibit even parities
under a space-reflection symmetry, the first four and the sequence of even
order charges are exactly conserved during the scattering process of the
solitons. We perform extensive numerical simulations and consider the bright
solitons with deformed potential $ V = \frac{ 2\eta}{2+ \epsilon} \(
|\psi|^2\)^{2 + \epsilon}, \epsilon \in \IR, \eta<0$. However, for two-soliton
field components without definite parity we also show numerically the vanishing
of the first non-trivial anomaly and the exact conservation of the relevant
charge. So, the parity symmetry seems to be a sufficient but not a necessary
condition for the existence of the infinite tower of conserved charges. The
model supports elastic scattering of solitons for a wide range of values of the
amplitudes and velocities and the set $\{\eta, \epsilon\}$. Since the NLS
equation is ubiquitous, our results may find potential applications in several
areas of non-linear science.
