修正的非线性薛定谔模型、${\cal C}{\cal P}_s{\cal T}_d$不变$N-$亮孤子和无限大反常电荷塔
Modified non-linear Schrödinger models, ${\cal C}{\cal P}_s{\cal T}_d$ invariant $N-$bright solitons and infinite towers of anomalous charges
摘要：
修正的非线性薛定谔模型（MNLS）$ i \partial_{t} ψ(x,t) + \partial^2_{x} ψ(x,t) - [\frac{δV}{δ|ψ|^2} ] ψ(x,t) = 0,$，其中$ψ\in C$和$V: R_{+} \rightarrow R$，被认为是。我们发现，MNLS模型具有无限塔的准守恒律的孤子型配置与一个特殊的复杂的共轭，移动宇称和延迟时间反转（${\cal C}{\cal P}_s{\cal T}_d$）对称性。对于${\cal C}{\cal P}_s{\cal T}_d$不变的$N-$亮孤子，即使在标准NLS模型中也会出现无限的反常电荷塔。真正的守恒电荷是通过某种反常抵消机制出现的。引入了修正AKNS系统（MAKNS）的对偶Riccati型赝势公式，揭示了新的反常守恒律的无穷塔。此外，无穷塔的精确非局部守恒律被发现在一个线性系统制定。我们的分析结果支持的数值模拟的$2-$明孤子散射的潜在$ V = - \frac{ 2η}{2+ ε} ( |ψ|^2 )^{2 + ε}, ε\in R, η>0$。我们的数值模拟显示了亮孤子的弹性散射的范围广泛的值的集合$\{η, ε\}$和各种振幅和相对速度。AKNS型系统是一个普遍存在的系统，因此，我们的研究结果在非线性物理的许多领域都有潜在的应用，如玻色-爱因斯坦凝聚、超导、孤子湍流以及规范理论、可积模型和引力理论之间的相互作用.
英文摘要：
Modifications of the non-linear Schrödinger model (MNLS) $ i \partial_{t} ψ(x,t) + \partial^2_{x} ψ(x,t) - [\frac{δV}{δ|ψ|^2} ] ψ(x,t) = 0,$ where $ψ\in C$ and $V: R_{+} \rightarrow R$, 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 (${\cal C}{\cal P}_s{\cal T}_d$) 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. 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{ 2η}{2+ ε} ( |ψ|^2 )^{2 + ε}, ε\in R, η>0$. Our numerical simulations show the elastic scattering of bright solitons for a wide range of values of the set $\{η, ε\}$ 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.
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