Integrable turbulence and statistical characteristics of chaotic wave field in the Kundu-Eckhaus equation

Integrable turbulence, as an irregular behavior in dynamic systems, has attracted a lot of attention in integrable and Hamiltonian systems. This article focuses on the studies of integrable turbulence phenomena of the Kundu-Eckhaus (KE) equation as well as the generation of rogue waves from the numerical and statistical viewpoints. First, via the Fourier collocation method, we obtain the spectral portraits of different analytical solutions. Second, we perform the numerical simulation on the KE equation under the initial condition of a plane wave with random noise to simulate the chaotic wave fields. Then, we analyze the influences of standard deviation and correlation length on the integrable turbulence and amplitude of wave field. It's found that both of the two parameters have positive effects on the generation probability of rogue wave caused by the interactions. But only the variation of standard deviation can lead to the transition from the breather turbulence to soliton turbulence. Furthermore, by analyzing the effects of additional higher-order nonlinear terms on the chaotic wave field, we find that those two higher-order nonlinear effects in the KE equation can lead to a larger amplitude of the chaotic wave field and a higher probability of generating rouge waves compared with the NLS equation.