Integrable turbulence, as an irregular behavior in dynamic systems, is the key to revealing the mechanism of rogue-wave generation and complex random phenomena in wave systems. In this paper we numerically and statistically investigate the integrable turbulence of a coupled nonlinear Schrödinger system with four-wave mixing effects, which describes optical wave propagation in multimode fibers. Different from prior studies of scalar integrable turbulence, our work deals with two coupled polarization components whose nonlinear evolution is modulated by a shared potential containing cross-phase modulation and four-wave mixing. It reveals that an increase in the initial perturbation standard deviation leads to a transition from breather turbulence to soliton turbulence, significantly enhancing the occurrence probability of rogue waves. This can be reflected in an enhanced heavy-tailed feature of the probability density function and an increase in the asymptotic values of the stable state for skewness and kurtosis. Moreover, the change of the correlation length mainly affects the ratio of solitons to breathers, with a relatively minor impact on the probability distribution. It also indicates that the four-wave mixing effect drives the redistribution of spectral energy, further modulating the generation dynamics of rogue waves. These findings deepen the understanding of integrable turbulence and rogue-wave generation in coupled nonlinear systems with four-wave mixing effects.

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• Mathematical physics
• Turbulent mixing