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Deformations of the focusing non-linear Schr\"odinger model (NLS) areconsidered in the context of the quasi-integrability concept. We strengthen theresults of JHEP09(2012)103 for bright soliton collisions. We addressed thefocusing NLS as a complement to the one in JHEP03(2016)005, in which themodified defocusing NLS models with dark solitons were shown to exhibit aninfinite tower of exactly conserved charges. We show, by means of analyticaland numerical methods, that for certain two-bright-soliton solutions, in whichthe modulus and phase of the complex modified NLS field exhibit even paritiesunder a space-reflection symmetry, the first four and the sequence of evenorder charges are exactly conserved during the scattering process of thesolitons. We perform extensive numerical simulations and consider the brightsolitons with deformed potential $ V = \frac{ 2\eta}{2+ \epsilon} $|\psi|^2$^{2 + \epsilon}, \epsilon \in \IR, \eta<0$. However, for two-solitonfield components without definite parity we also show numerically the vanishingof the first non-trivial anomaly and the exact conservation of the relevantcharge. So, the parity symmetry seems to be a sufficient but not a necessarycondition for the existence of the infinite tower of conserved charges. Themodel supports elastic scattering of solitons for a wide range of values of theamplitudes and velocities and the set $\{\eta, \epsilon\}$. Since the NLSequation is ubiquitous, our results may find potential applications in severalareas of non-linear science.

Quasi-integrable non-linear Schr\"odinger models, infinite towers of exactly conserved charges and bright solitons