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

Abstract

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 $\mathcal{C}{\mathcal{P}}_s{\mathcal{T}}_d$ 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 $\epsilon$, such that the standard NLS is recovered for $\epsilon =0$. Here we present the numerical simulations for small values of $|\epsilon |$, 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.

12pages, 10 figures. NODYCON2021 conference

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