Title: Asymmetric integrable turbulence and rogue wave statistics for the derivative nonlinear Schrödinger equation Show Chinese title

Title: 非对称可积湍流和导数非线性薛定谔方程的 rogue 波统计特性

Abstract: 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. Show Chinese abstract

Abstract: 我们研究了DNLS方程中平面波的调制不稳定性（MI）所产生的非对称可积湍流和异常波（RWs）。 \(n\)阶矩和系综平均的动能和势能表现出向稳态值振荡收敛。 具体而言，这些指标的振幅随时间以\(t^{-1.36}\)的方式渐近衰减，而相位偏移则以\(t^{-0.78}\)的速率呈现非线性衰减。 这些振荡的频率被观察到是MI最大增长率的两倍。 这些振荡可以分为两种不同的类型：一种与系综平均势能模$|\langle H_4\rangle|$同相，另一种则为反相。 同时，这种统一性也体现在给定\( k \)的波作用谱\( S_k(t) \)、给定\( x \)的自相关函数\( g(x,t) \)以及概率密度函数\( P(I,t) \)上。 湍流的关键特征是波作用谱，它遵循幂律分布\( |k+3|^{-\alpha} \)，除了$k=-3$。 与NLS方程不同，DNLS设置中的湍流是不对称的，主要是由于来自MI的平面波的波数与扰动波数之间的不对称性。当\( S_k \)的渐近峰值出现在\( k = -3 \)时，自相关函数在\( x \to \pm L/2 \)处表现出非零水平。波强度的PDF以振荡方式渐近接近指数分布。然而，在非线性相位的初始阶段，MI略微增加了RWs的发生率。这发生在势模处于最小值的时刻，此时在\( I\in [12, 15] \)范围内的RWs发生概率明显高于渐近稳态情况。