Statistical properties and rogue waves in integrable turbulence for the modified nonlinear Schrödinger equation with self-steepening effect

In this study, we numerically investigate the integrable turbulence for the modified nonlinear Schrödinger (MNLS) equation with self-steepening effect. We focus on the statistical metrics of nonlinear wave turbulence excited from a constant background with random perturbation. It's found that higher values of parameters in the initial condition can lead to greater asymptotic kurtosis and heavy-tailed probability density functions, signaling an elevated likelihood for the occurrence of rogue waves. Meanwhile, by analyzing the autocorrelation function, we reveal the robustness of the quasi-periodicity in the modulation instability process. Moreover, the physical spectrum is characterized by the gradual disappearance of the “onion dome” shape and tail elevation, which reflect a shift from breather turbulence to soliton turbulence. Different from the standard NLS equation, the self-steepening effect will break the symmetry of physical spectrum and suppress the generation of rogue waves. Finally, we investigate the distribution of IST eigenvalue spectrum in the integrable turbulence.