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

Abstract

Modifications of the non-linear Schrödinger model (MNLS) $i{\partial }_t\psi (x,t)+{\partial }_x^2\psi (x,t)-[\frac{\delta V}{\delta |\psi {|}^2}]\psi (x,t)=0,$ where $\psi \in C$ and $V:R_{+}\to R$, are considered. We show that the MNLS models possess infinite towers of quasi-conservation laws for soliton-type configurations with a special complex conjugation, shifted parity and delayed time reversion ($\mathcal{C}{\mathcal{P}}_s{\mathcal{T}}_d$) symmetry. Infinite towers of anomalous charges appear even in the standard NLS model for $\mathcal{C}{\mathcal{P}}_s{\mathcal{T}}_d$ invariant $N-$bright solitons. The true conserved charges emerge through some kind of anomaly cancellation mechanism. A dual Riccati-type pseudo-potential formulation is introduced for a modified AKNS system (MAKNS) and infinite towers of novel anomalous conservation laws are uncovered. In addition, infinite towers of exact non-local conservation laws are uncovered in a linear system formulation. 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.

arXiv comment58 pages, 15 figures. A new section added with derivations of novel anomalous charges in the pseudo-potential approach. A recent Ref. (arXiv:2102.07190 [hep-th]) about similar results on dark-soliton simulations has been added

References in corpus (12)

• Dark solitons in atomic Bose-Einstein condensates: from theory to experiments
• Experimental evidence of a hydrodynamic soliton gas
• Integrable generalizations of Schrodinger maps and Heisenberg spin models from Hamiltonian flows of curves and surfaces
• The concept of quasi-integrability: a concrete example
• Bistability and instability of dark-antidark solitons in the cubic-quintic nonlinear Schroedinger equation
• The concept of quasi-integrability for modified non-linear Schrodinger models
• Coverings and fundamental algebras for partial differential equations
• Quasi-Integrability in Supersymmetric Sine-Gordon Models
• Vector NLS hierarchy solitons revisited: dressing transformation and tau function approach
• Gravity Dual of Two-Dimensional $\mathcal{N}=(2,2{)}^{\ast }$ Supersymmetric Yang-Mills Theory and Integrable Models
• Modified non-linear Schrödinger models, $\mathcal{C}{\mathcal{P}}_s{\mathcal{T}}_d$ symmetry, dark solitons and infinite towers of anomalous charges
• Riccati-type pseudopotentials, conservation laws and solitons of deformed sine-Gordon models

Cited by in corpus (1)

• Modified AKNS model, Riccati-type pseudo-potential approach and infinite towers of quasi-conservation laws