Asymmetric integrable turbulence and rogue wave statistics for the derivative nonlinear Schr\"odinger equation

We investigate the asymmetric integrable turbulence and rogue waves (RWs) emerging from the modulation instability (MI) of plane waves for the DNLS equation. The $n$-th moments and ensemble-averaged kinetic and potential energy exhibit oscillatory convergence towards their steady-state values. Specifically, the amplitudes of oscillations for these indexes decay asymptotically with time as $t^{-1.36}$, while the phase shifts demonstrate a nonlinear decay with a rate of $t^{-0.78}$. The frequency of these oscillations is observed to be twice the maximum growth rate of MI. These oscillations can be classified into two distinct types: one is in phase with ensemble-averaged potential energy modulus $|\langle H_4\rangle|$, and the other is anti-phase. At the same time, this unity is also reflected in the wave-action spectrum $S_k(t)$ for a given $k$, the auto-correlation function $g(x,t)$ for a given $x$, as well as the PDF $P(I,t)$. The critical feature of the turbulence is the wave-action spectrum, which follows a power-law distribution of $|k+3|^{-\alpha}$ expect for $k=-3$. Unlike the NLS equation, the turbulence in the DNLS setting is asymmetric, primarily due to the asymmetry between the wave number of the plane wave from the MI and the perturbation wave number.. As the asymptotic peak value of $S_k$ is observed at $k = -3$, the auto-correlation function exhibits a nonzero level as $x \to \pm L/2$. The PDF of the wave intensity asymptotically approaches the exponential distribution in an oscillatory manner. However, during the initial stage of the nonlinear phase, MI slightly increases the occurrence of RWs. This happens at the moments when the potential modulus is at its minimum, where the probability of RWs occurring in the range of $I\in [12, 15]$ is significantly higher than in the asymptotic steady state.