Nonlinear dispersion relation in integrable turbulence.

We investigate numerically and experimentally the concept of nonlinear dispersion relation (NDR) in the context of partially coherent waves propagating in a one-dimensional water tank. The nonlinear random waves have a narrow-bandwidth Fourier spectrum and are described at leading order by the one-dimensional nonlinear Schrödinger equation. The problem is considered in the framework of integrable turbulence in which solitons play a key role. By using a limited number of wave gauges, we accurately measure the NDR of the slowly varying envelope of the deep-water waves. This enables the precise characterization of the frequency shift and the broadening of the NDR while also revealing the presence of solitons. Moreover, our analysis shows that the shape and the broadening of the NDR provides signatures of the deviation from integrable turbulence that is induced by high order effects in experiments. We also compare our experimental observations with numerical simulations of Dysthe and of Euler equations.

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

Soliton gases represent large random soliton ensembles in physical systems that exhibit 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 a 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.

Collective strategies and cyclic dominance in asymmetric predator-prey spatial games.

Predators may attack isolated or grouped prey in a cooperative, collective way. Whether a gregarious behavior is advantageous to each species depends on several conditions and game theory is a useful tool to deal with such a problem. We here extend the Lett et al. (2004) to spatially distributed populations and compare the resulting behavior with their mean-field predictions for the coevolving densities of predator and prey strategies. Besides its richer behavior in the presence of spatial organization, we also show that the coexistence phase in which collective and individual strategies for each group are present is stable because of an effective, cyclic dominance mechanism similar to a well-studied generalization of the Rock-Paper-Scissors game with four species, a further example of how ubiquitous this coexistence mechanism is.

Experimental study of three-wave interactions among capillary-gravity surface waves.

In propagating wave systems, three- or four-wave resonant interactions constitute a classical nonlinear mechanism exchanging energy between the different scales. Here we investigate three-wave interactions for gravity-capillary surface waves in a closed laboratory tank. We generate two crossing wave trains and we study their interaction. Using two optical methods, a local one (laser doppler vibrometry) and a spatiotemporal one (diffusive light photography), a third wave of smaller amplitude is detected, verifying the three-wave resonance conditions in frequency and in wave number. Furthermore, by focusing on the stationary regime and by taking into account viscous dissipation, we directly estimate the growth rate of the resonant mode. The latter is then compared to the predictions of the weakly nonlinear triadic resonance interaction theory. The obtained results confirm qualitatively and extend previous experimental results obtained only for collinear wave trains. Finally, we discuss the relevance of three-wave interaction mechanisms in recent experiments studying gravity-capillary turbulence.