We use the spectral kinetic theory of soliton gas to investigate the likelihood of extreme events in integrable turbulence described by the one-dimensional focusing nonlinear Schrödinger equation (fNLSE). This is done by invoking a stochastic interpretation of the inverse scattering transform for fNLSE and analytically evaluating the kurtosis of the emerging random nonlinear wave field in terms of the spectral density of states of the corresponding soliton gas. We then apply the general result to two fundamental scenarios of the generation of integrable turbulence: (i) the asymptotic development of the spontaneous modulational instability of a plane wave, and (ii) the long-time evolution of strongly nonlinear, partially coherent waves. In both cases, involving the bound state soliton gas dynamics, the analytically obtained values of the kurtosis are in perfect agreement with those inferred from direct numerical simulations of the fNLSE, providing the long-awaited theoretical explanation of the respective rogue wave statistics. Additionally, the evolution of a particular nonbound state gas is considered, providing important insights related to the validity of the so-called virial theorem.

中文翻译：

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可积湍流中的极端事件统计

我们使用孤子气体的谱动力学理论来研究由一维聚焦非线性薛定谔方程（fNLSE）描述的可积湍流中极端事件的可能性。这是通过调用 fNLSE 的逆散射变换的随机解释并根据相应孤子气体的态谱密度分析评估新出现的随机非线性波场的峰度来完成的。然后，我们将一般结果应用于可积湍流生成的两个基本场景：（i）平面波自发调制不稳定性的渐近发展，以及（ii）强非线性、部分相干波的长期演化。在这两种涉及束缚态孤子气体动力学的情况下，分析获得的峰度值与 fNLSE 的直接数值模拟推断出的值完全一致，为各自的异常波统计提供了期待已久的理论解释。此外，还考虑了特定非束缚态气体的演化，提供了与所谓维里定理的有效性相关的重要见解。