Compressed Exponential and Dual Structures in Correlation Functions of Weak Turbulence due to a Nambu–Goldstone Mode

In this study, we investigate the Nikolaevskii equation, which develops turbulence via supercritical bifurcation. The statistical properties of the turbulence governed by the equation are examined through the total autocorrelation function. The total autocorrelation function fits the Kohlrausch–Williams–Watts expression. By increasing the control parameter, we demonstrate a transition from compressed to stretched exponential behavior for the autocorrelation function of Nikolaevskii turbulence. Furthermore, our study finds that the modal correlation function of Nikolaevskii turbulence consists of dual structures: algebraic and exponential decay, with two types of characteristic exponents existing in the exponential decay regime. The transition from compressed to stretched exponential in the total correlation function is related to the change in these characteristic exponents.