具有增益和损耗的非线性薛定谔模型的时空代数定域波形
Spatiotemporal algebraically localized waveforms for a nonlinear Schrödinger model with gain and loss
摘要：
本文研究了一个包含线性和非线性增益/损失的非线性薛定谔（NLS）模型解的渐近性态。首先，我们分析描述的动力学制度（取决于增益/损失的强度），有限时间崩溃，衰减，和全球存在的解决方案的动态。然后，对于所有上述参数制度，我们使用直接数值模拟来研究对应于代数衰减的初始数据的动力学。我们确定消失的初始条件的动态之间的关键差异，收敛到一个有限的常数背景：在前（后）种情况下，我们发现强（弱）崩溃或衰减，当增益/损失参数从相关的制度。我们的主要结果之一，是在所有上述制度，非零的初始数据过渡时空，代数衰减波形。虽然系统是不可积的，这些波形的演变是令人想起的可积NLS极限的游隼流氓波的演变。增益和损失的参数范围内，这种现象持续也触及。
英文摘要：
We consider the asymptotic behavior of the solutions of a nonlinear Schrödinger (NLS) model incorporating linear and nonlinear gain/loss. First, we describe analytically the dynamical regimes (depending on the gain/loss strengths), for finite-time collapse, decay, and global existence of solutions in the dynamics. Then, for all the above parametric regimes, we use direct numerical simulations to study the dynamics corresponding to algebraically decaying initial data. We identify crucial differences between the dynamics of vanishing initial conditions, and those converging to a finite constant background: in the former (latter) case we find strong (weak) collapse or decay, when the gain/loss parameters are selected from the relevant regimes. One of our main results, is that in all the above regimes, non-vanishing initial data transition through spatiotemporal, algebraically decaying waveforms. While the system is nonintegrable, the evolution of these waveforms is reminiscent to the evolution of the Peregrine rogue wave of the integrable NLS limit. The parametric range of gain and loss for which this phenomenology persists is also touched upon.
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