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Modified non-linear Schrödinger models, ${\cal C}{\cal P}_s{\cal T}_d$ symmetry, dark solitons and infinite towers of anomalous charges

Some modified (defocusing) non-linear Schrödinger models (MNLS) possess infinite towers of anomalous conservation laws with asymptotically conserved charges. The so-called anomalies of the quasi-conservation laws vanish upon space-time integration for a special ${\cal C}{\cal P}_s{\cal T}_d$ symmetric field configurations. We verify numerically the degree of modifications of the charges around the dark-soliton interaction regions by computing numerically some representative anomalies related to lowest order quasi-conservation laws of the non-integrable cubic-quintic NLS model as a modified (defocusing) NLS model. This modification depends on the parameter $ε$, such that the standard NLS is recovered for $ε=0$. Here we present the numerical simulations for small values of $|ε|$, and show that the collision of two dark solitons are elastic. The NLS-type equations are quite ubiquitous in several areas of non-linear science.