We propose a new approach to study the long time dynamics of the wave kinetic equation in the statistical description of acoustic turbulence. The approach is based on rewriting the discrete version of the wave kinetic equation in the form of a chemical reaction network, then employing techniques used to study the Global Attractor Conjecture to investigate the long time dynamics of the newly obtained chemical system. We show that the solution of the chemical system converges to an equilibrium exponentially in time. In addition, a resonance broadening modification of the acoustic wave kinetic equation is also studied with the same technique. For the near-resonance equation, if the resonance broadening frequency is larger than a threshold, the solution of the system goes to infinity as time evolves.

中文翻译：

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声波湍流理论的反应网络方法

我们提出了一种新方法来研究声湍流统计描述中波动动力学方程的长时间动力学。该方法基于以化学反应网络的形式重写波动动力学方程的离散版本，然后采用用于研究全局吸引子猜想的技术来研究新获得的化学系统的长时间动力学。我们表明，化学系统的解决方案在时间上呈指数收敛。此外，还使用相同的技术研究了声波动力学方程的共振展宽修正。对于近共振方程，如果共振展宽频率大于阈值，则系统的解随着时间的推移趋于无穷大。