Statistics of extreme events in integrable turbulence

Abstract

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 (noise induced) 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 the fNLSE, providing the long-awaited theoretical explanation of the respective rogue wave statistics. Additionally, the evolution of a particular non-bound state gas is considered providing important insights related to the validity of the so-called virial theorem.

13 pages, 5 figures

References in corpus (6)

• Rogue waves and analogies in optics and oceanography
• Optical Rogue Waves in integrable turbulence
• Soliton Turbulence in Shallow Water Ocean Surface Waves
• Experimental evidence of a hydrodynamic soliton gas
• Spontaneous emergence of rogue waves in partially coherent waves: a quantitative experimental comparison between hydrodynamics and optics
• Recent developments in spectral theory of the focusing NLS soliton and breather gases: the thermodynamic limit of average densities, fluxes and certain meromorphic differentials; periodic gases

Cited by in corpus (5)

• Asymmetric integrable turbulence and rogue wave statistics for the derivative nonlinear Schrödinger equation
• Observation of a generalized Gibbs ensemble in photonics
• One-dimensional wave kinetic theory
• Evolution of internal cnoidal waves with local defects in a two-layer fluid with rotation
• Soliton Condensates for the Focusing Nonlinear Schrödinger Equation: a Non-Bound State Case