Speaker
Description
Modifications of the non-linear Schr\"odinger model (MNLS) $ i \partial_{t} \psi(x,t) + \partial^2_{x} \psi(x,t) - [\frac{\delta V}{\delta |\psi|^2} ] \psi(x,t) = 0,$ where $\psi \in C$ and $V: R_{+} \rightarrow R$, are considered. We show that the quasi-integrable MNLS models possess infinite towers of quasi-conservation laws for soliton-type configurations with a special complex conjugation, shifted parity and delayed time reversion (${\cal C}{\cal P}_s{\cal T}_d$) symmetry. Infinite towers of anomalous charges appear even in the standard NLS model for ${\cal C}{\cal P}_s{\cal T}_d$ invariant $N-$bright solitons. The true conserved charges emerge through some kind of anomaly cancellation mechanism, since a convenient linear combination of the relevant anomalies vanishes.
A Riccati-type pseudo-potential is introduced for a modified AKNS system (MAKNS), which reproduces the MNLS quantities upon a reduction process. Two infinite towers of exact non-local conservation laws are uncovered in this framework.
Our analytical results are supported by numerical simulations of $2-$bright-soliton scatterings with potential $ V = - \frac{ 2\eta}{2+ \epsilon} ( |\psi|^2 )^{2 + \epsilon}, \epsilon \in R, \eta>0$. Our numerical simulations show the elastic scattering of bright solitons for a wide range of values of the set $\{\eta, \epsilon\}$ and a variety of amplitudes and relative velocities. The AKNS-type system is quite ubiquitous, and so, our results may find potential applications in several areas of non-linear physics, such as Bose-Einstein condensation, superconductivity, soliton turbulence and the triality among gauge theories, integrable models and gravity theories.
