S. Aumaître

**Abstract: We study experimentally, in a
large-scale basin, the propagation of unidirectional deep water
gravity waves stochastically modulated in phase. We observe the
emergence of nonlinear localized structures that evolve on a
stochastic wave background. Such a coexistence is expected by
the integrable turbulence theory for the nonlinear Schrödinger
equation (NLSE), and we report the first experimental
observation in the context of hydrodynamic waves. We
characterize the formation, the properties and the dynamics of
these nonlinear coherent structures (solitons and extreme
events) within the incoherent wave background. The extreme
events result from the strong steepening of wave train fronts,
and their emergence occurs after roughly one nonlinear length
scale of propagation (estimated from NLSE). Solitons arise when
nonlinearity and dispersion are weak, and of the same order of
magnitude as expected from NLSE. We characterize the statistical
properties of this state. The number of solitons and extreme
events is found to increase all along the propagation, the
wave-field distribution has a heavy tail, and the surface
elevation spectrum is found to scale as a frequency power-law
with an exponent -4.5 ± 0.5. Most of these observations are
compatible with the integrable turbulence theory for NLSE
although some deviations (e.g. power-law spectrum, asymmetrical
extreme events) result from effects proper to hydrodynamic
waves.
**