具有时空扰动的非线性薛定谔方程
Nonlinear Schrödinger Equation with Spatio-Temporal Perturbations
AI总结：
本文研究带时空驱动、阻尼和线性稳定项的三次非线性薛定谔方程中孤子的动力学，建立集体坐标理论并解析、数值求解，发现孤子平均单向运动，并提出由动量-速度曲线负斜率分支判断不稳定性和估计孤子寿命的猜想，模拟结果完全证实该猜想。
摘要：
我们研究了带以下扰动的三次非线性薛定谔方程（NLSE）中孤子的动力学：形式为 $f(x,t) = a \exp[i K(t) x]$ 的非参数时空驱动、阻尼，以及一个用于稳定被驱动孤子的线性项。利用范数、动量和能量的时间演化，或者采用拉格朗日方法，我们建立了一种集体坐标理论，该理论为我们的四个集体坐标给出一组常微分方程（ODE）。对于恒定的、空间周期力 $f(x)$ 的情形，我们对这些常微分方程进行了解析和数值求解。孤子位置表现出围绕具有恒定速度的平均轨迹的振荡。这意味着尽管力的空间平均值为零，孤子平均而言仍做单向运动。振荡的幅度远小于 $f(x)$ 的周期。为了确定上述解在哪些区域是稳定的，我们计算了孤子动量 $P(t)$ 和孤子速度 $V(t)$ 的时间演化：这是曲线 $P(V)$ 的参数表示，孤子随时间演化时会经过该曲线。我们的猜想是，如果该曲线具有负斜率的分支，孤子将变得不稳定。这一猜想被我们对受扰 NLSE 的模拟完全证实。此外，该曲线还能很好地估计孤子寿命：负斜率分支越短，孤子存活时间越长。
英文摘要：
We investigate the dynamics of solitons of the cubic Nonlinear Schrödinger Equation (NLSE) with the following perturbations: non-parametric spatio-temporal driving of the form $f(x,t) = a \exp[i K(t) x]$, damping, and a linear term which serves to stabilize the driven soliton. Using the time evolution of norm, momentum and energy, or, alternatively, a Lagrangian approach, we develop a Collective-Coordinate-Theory which yields a set of ODEs for our four collective coordinates. These ODEs are solved analytically and numerically for the case of a constant, spatially periodic force $f(x)$. The soliton position exhibits oscillations around a mean trajectory with constant velocity. This means that the soliton performs, on the average, a unidirectional motion although the spatial average of the force vanishes. The amplitude of the oscillations is much smaller than the period of $f(x)$. In order to find out for which regions the above solutions are stable, we calculate the time evolution of the soliton momentum $P(t)$ and soliton velocity $V(t)$: This is a parameter representation of a curve $P(V)$ which is visited by the soliton while time evolves. Our conjecture is that the soliton becomes unstable, if this curve has a branch with negative slope. This conjecture is fully confirmed by our simulations for the perturbed NLSE. Moreover, this curve also yields a good estimate for the soliton lifetime: the soliton lives longer, the shorter the branch with negative slope is.
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