Asymmetric integrable turbulence and rogue wave statistics for the derivative nonlinear Schrödinger equation

TL;DR

The paper addresses how modulation instability of a plane wave in the derivative nonlinear Schrödinger equation (DNLS) gives rise to asymmetric integrable turbulence and rogue waves. Using extensive long-time simulations, it analyzes moments, ensemble energies, wave-action spectra, PDFs, and spatial correlations to reveal two-phase, oscillatory approach to steady states with amplitudes decaying as $t^{-1.36}$ and nonlinear phase shifts decaying as $t^{-0.78}$, with oscillation frequency twice the MI growth rate. The wave-action spectrum shows a central power-law region $S_k(t) o C|k+3|^{-oldsymbol{ m α}}$ around $k=-3$, reflecting DNLS-specific anisotropy, while the intensity PDF approaches an exponential form and higher moments converge to Rayleigh predictions, indicating Gaussian-like asymptotic turbulence. These results establish a distinct, asymmetric IT regime in the DNLS and provide quantitative benchmarks for spectral, statistical, and correlation measures in integrable turbulence. The findings have implications for understanding RW statistics and turbulence in other integrable systems and motivate future exploration of higher-dimensional or alternative initial-condition scenarios.

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 \( |k+3|^{-α} \) 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.

Figures (12)

• Figure 1: A typical evolution of the turbulent wave field $|\psi(x,t)|$ for a single realization corresponds to the parameters at MI region (a) with $k = - 3$, and at MS region (b) with $k = -1/4$.
• Figure 2: The largest RWs detected in the realization corresponding to figure \ref{['MI']}(a). (a1) The comparison between the largest RW detected in the realization and the cross-profile of exact RW solution, where the blue solid and red dashed lines represent the numerical and exact ones, respectively. The space-time representation near the location where the maximum amplitude is attained of numerical realization at (a2) and exact solution at (a3). To enhance visualization, we translate the point of maximum amplitude to the coordinates $(x,t) = (0,0)$.
• Figure 3: (a) The PDF of intensity $I=|\psi|^2$, where the blue solid line represents asymptotic state of IT, while the red dashed line denotes the exponential distribution $\exp(-I)$. (b) The asymptotic values of the moments $M^{(n)}_A$, where $n = 1, \dots, 10$, are represented by blue circles. These values are compared with the Rayleigh prediction, $\Gamma\left(\frac{n}{2} + 1\right)$, which is shown as a dashed red line.
• Figure 4: (a) The evolution of $n$-th moment $\left[M^{(n)}(t)\right]^{1/n}$ (see Eq. (\ref{['Moment']})) ($n=1,2,3$) with time $t$. (b) The evolution of ensemble-averaged kinetic energy $\langle H_k(t) \rangle$ Eq. (\ref{['KE']}) and potentia energy $\langle H_4(t)\rangle$ Eq. (\ref{['PE']}) with time $t$.
• Figure 5: (a1) The magnitude of the deviations of the extreme of $M^{(1)}(t)$ from its asymptotic value $M_A^{(1)}$, as a function of time $t$. (a2) The evolution of the moment $M^{(1)}(t)$ (solid black line) is fitted by the function $M_A^{(1)} + \frac{a_1}{t^{\alpha_1}} \sin \left(b_1 t+c_1/t^{\beta_1}+\theta_0^{(1)}\right)$, with the parameters $M_{A}^{1}\approx0.886$, $\{a_1,\alpha_1,b_1,c_1,\beta_1,\theta_0^{(1)}\}\approx$$\{0.89,1.36,4.89,51.41,0.78,-44.59\}$ (dashed red line). (b1) The same as (a1), expect $M^{(1)}(t)$ is changed into $M^{(4)}(t)$. (b2) The same as (a2), expect $M^{(1)}(t)$ is changed into $M^{(4)}(t)$, as well as the parameters $M_{A}^{4}\approx2, \{a_2,\alpha_2,b_2,c_2,\beta_2,\theta_0^{(2)}\}\approx\{9.39,1.35,4.89,51.06,0.80,-41.46\}.$