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The concept of quasi-integrability has been examined in the context ofdeformations of the defocusing non-linear Schr\"odinger model (NLS). Ourresults show that the quasi-integrability concept, recently discussed in thecontext of deformations of the sine-Gordon, Bullough-Dodd and focusing NLSmodels, holds for the modified defocusing NLS model with dark soliton solutionsand it exhibits the new feature of an infinite sequence of alternatingconserved and asymptotically conserved charges. For the special case of twodark soliton solutions, where the field components are eigenstates of aspace-reflection symmetry, the first four and the sequence of even ordercharges are exactly conserved in the scattering process of the solitons. Suchresults are obtained through analytical and numerical methods, and employadaptations of algebraic techniques used in integrable field theories. Weperform extensive numerical simulations and consider the scattering of darksolitons for the cubic-quintic NLS model with potential $V =\eta I^2 -\frac{\epsilon}{6} I^3 $ and the saturable type potential satisfying $V'[I] =2\eta I - \frac{\epsilon I^q}{1+ I^q},\,q \in \IZ_{+}$, with a deformationparameter $\epsilon \in \IR$ and $I=|\psi|^2$. The issue of the renormalizationof the charges and anomalies, and their (quasi)conservation laws are properlyaddressed. The saturable NLS supports elastic scattering of two solitonsolutions for a wide range of values of $\{\eta, \epsilon, n\}$. Our resultsmay find potential applications in several areas of non-linear science, such asthe Bose-Einstein condensation.

Quasi-integrability in the modified defocusing non-linear Schr\"odinger model and dark solitons