五次非线性薛定谔方程的强坍缩Turbulence
Strong Collapse Turbulence in Quintic Nonlinear Schrödinger Equation
摘要：
研究了具有强迫项和线性耗散项及非线性耗散项的五次一维非线性薛定谔方程.五次非线性导致在空间和时间上随机分布的多个塌缩事件，形成强迫湍流。如果没有耗散，这些坍缩中的每一个都会产生有限时间奇异性，但耗散项会阻止奇异性的实际形成。在发展湍流的统计稳定状态下，空间关联函数具有普遍的形式，关联长度由调制不稳定尺度决定。该尺度下的振幅波动接近高斯，而概率密度函数（PDF）的大振幅尾部是强非高斯的，具有类似幂的行为。小振幅的近高斯波动为大坍缩事件的形成埋下了种子。这些事件的普遍时空形式与其最大振幅的PDF一起定义了大振幅波动的PDF的类似幂的尾部，即，强湍流的不稳定性。
英文摘要：
We consider the quintic one dimensional nonlinear Schrödinger equation with forcing and both linear and nonlinear dissipation. Quintic nonlinearity results in multiple collapse events randomly distributed in space and time forming forced turbulence. Without dissipation each of these collapses produces finite time singularity but dissipative terms prevents actual formation of singularity. In statistical steady state of the developed turbulence the spatial correlation function has a universal form with the correlation length determined by the modulational instability scale. The amplitude fluctuations at that scale are nearly-Gaussian while the large amplitude tail of probability density function (PDF) is strongly non-Gaussian with power-like behavior. The small amplitude nearly-Gaussian fluctuations seed formation of large collapse events. The universal spatio-temporal form of these events together with the PDF for their maximum amplitudes define the power-like tail of PDF for large amplitude fluctuations, i.e., the intermittency of strong turbulence.
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