Nonlinear Spectral Synthesis of Soliton Gas in Deep-Water Surface Gravity Waves

Abstract

Soliton gases represent large random soliton ensembles in physical systems that display integrable dynamics at the leading order. Despite significant theoretical developments and observational evidence of ubiquity of soliton gases in fluids and optical media their controlled experimental realization has been missing. We report the controlled synthesis of a dense soliton gas in deep-water surface gravity waves using the tools of nonlinear spectral theory (inverse scattering transform (IST)) for the one-dimensional focusing nonlinear Schrödinger equation. The soliton gas is experimentally generated in a one-dimensional water tank where we demonstrate that we can control and measure the density of states, i. e. the probability density function parametrizing the soliton gas in the IST spectral phase space. Nonlinear spectral analysis of the generated hydrodynamic soliton gas reveals that the density of states slowly changes under the influence of perturbative higher-order effects that break the integrability of the wave dynamics.

Experiments in Surface Gravity–Capillary Wave Turbulence

“…A transition has also been observed from a wave turbulence regime to a solitonic (nondispersive) regime when the forcing increases at weak depth (Hassaini & Mordant 2017). This so-called "soliton gas" regime is predicted by integrable turbulence (Zakharov 1971), a state basically different from wave turbulence that involves coherent structures (solitons) within stochastic waves (Cazaubiel et al 2018, Redor et al 2019, 2020, Suret et al 2020. Rather than decreasing the fluid depth, the transition from a dispersive to a nondispersive regime of hydrodynamic wave turbulence can be reached using a magnetic fluid (ferrofluid) within a magnetic field (Boyer & Falcon 2008, Dorbolo & Falcon 2011, Ricard & Falcon 2021b).…”

Prediction and manipulation of hydrodynamic rogue waves via nonlinear spectral engineering

“…Color gradient represents the length of the nonlinear propagation and varies from blue (z = 6 m) to red (z = 120 m). When the chirp parameter is small the discrete IST spectrum is almost conserved for both the experiments and the Dysthe model [54], which signifies that the nonlinear evolution is close to the one described by the 1-D fNLSe. Indeed, for µ = −0.8 the gradient catastrophe occurs beyond the water tank length so the intensity-related higher-order nonlinear effects can be neglected.…”

Soliton gas in integrable dispersive hydrodynamics

“…It has transpired later that the perspective of dispersive hydrodynamics, particularly the Whitham theory of modulated spectral finite-gap solutions provides an appealing mathematical framework for the generalisation of Zakharov's kinetic equation to the case of strongly interacting, dense soliton gas [40], [36], [42]. It has also been realised that soliton gas represents a ubiquitous physical phenomenon, a kind of strongly nonlinear turbulent wave motion, that can observed in the environmental conditions [23], [97] and realised in laboratory experiments [105], [120]. The recently discovered connections of the soliton gas dynamics and statistics with the topical areas of modulational instability and rogue wave formation [51], [53], [67] have provided the soliton gas research with further relevance.…”

“…In modulationally unstable media a semiclassical scenario of the transition to integrable soliton turbulence via a chain of topological bifurcations of the finite-gap spectra was theoretically proposed in [38] and to some extent realised in an optics experiment reported in [83]. Of particular relevance is the IST based approach to the controlled (as opposed to the spontaneous) experimental realisation of soliton gas developed in [120]. This approach enables the generation of a soliton gas with a prescribed DOS, which is crucial for the verification and utilisation of the spectral kinetic theory.…”

“…The observations and analysis of irregular soliton complexes in the ocean have been reported in [23], [116]. Recent laboratory experiments on the generation of shallow-water and deep water soliton gases were reported in [105] and [120] respectively. It has also been demonstrated that the soliton gas dynamics in the focusing NLS equation enables a remarkably accurate description of the statistical properties of the nonlinear stage of noise-induced modulational instability [51] as well as provides important insightsinto the dynamical and statistical mechanisms of the spontaneous formation of rogue waves [53], [5].…”