Integrable turbulence and statistical properties of chaotic wave fields in the complex modified Korteweg-de Vries equation
This paper investigates integrable turbulence and the mechanism for generating rogue waves in the complex modified Korteweg--de Vries (mKdV) equation through numerical simulations and statistical analysis. We solve the Lax pair eigenvalue problem using the Fourier collocation method and establish a correspondence among solitons, breathers, and rogue waves on the spectral plane. We generate chaotic wave fields by simulating the complex mKdV equation using the split-step Fourier method, starting from a plane wave with superimposed random noise. We analyze the effects of the standard deviation \(μ\) and the correlation length \(L_c\). The results show that increasing \(μ\) drives a transition from breather turbulence to soliton turbulence, increasing the probability of rogue-wave generation.In contrast, \(L_c\) primarily influences the spatial scale of the structures. As the constant background of the complex mKdV equation is modulationally stable, its integrable turbulence does not depend on modulational instability. Compared with the nonlinear Schrödinger (NLS) equation under identical initial conditions, the complex mKdV equation attains comparable amplitudes under strong perturbations, but exhibits a higher spatial gradient. This suggests that third-order dispersion and derivative nonlinearity alter the excitation mechanism and steepen the waveform.