离散湍流方程的大周期极限
The large-period limit for equations of discrete turbulence
摘要：
考虑大周期$L$环面上的阻尼/驱动三次NLS方程，该方程具有尺寸为$λ$的小的非线性，即适当尺度的随机强迫和耗散.我们检查其解决方案下的后续限制时，第一$λ\to 0$，然后$L\to \infty$。第一个极限，称为离散湍流的极限，是已知存在的，在这项工作中，我们研究的第二个极限$L\to\infty$离散湍流方程的解决方案。即把解按振幅分解为形式级数，并研究了该级数的二阶截断。我们证明了截断解的能量谱变得接近阻尼/驱动非线性波动方程的解。后者的动力学非线性类似于通常出现在波浪湍流的作品，但不同于它（特别是，它是非自治的）。除了从分析和随机分析的工具，我们的工作使用两个强大的结果从数论。
英文摘要：
We consider the damped/driven cubic NLS equation on the torus of a large period $L$ with a small nonlinearity of size $λ$, a properly scaled random forcing and dissipation. We examine its solutions under the subsequent limit when first $λ\to 0$ and then $L\to \infty$. The first limit, called the limit of discrete turbulence, is known to exist, and in this work we study the second limit $L\to\infty$ for solutions to the equations of discrete turbulence. Namely, we decompose the solutions to formal series in amplitude and study the second order truncation of this series. We prove that the energy spectrum of the truncated solutions becomes close to solutions of a damped/driven nonlinear wave kinetic equation. Kinetic nonlinearity of the latter is similar to that which usually appears in works on wave turbulence, but is different from it (in particular, it is non-autonomous). Apart from tools from analysis and stochastic analysis, our work uses two powerful results from the number theory.
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