Synchronization of Chaos and the Transition to wave turbulence

The onset of turbulence is one of the outstanding problems of theoretical physics [Landau & Lifshitz, 1987; Frisch, 1995]. A seminal contribution to this study was brought about by Ruelle and Takens, who described the onset of turbulence as a sequence of Hopf bifurcations yielding a chaotic attractor in the system phase space [Ruelle & Takens, 1971; Newhouse et al., 1978]. In spite of the recent progress in turbulence theory, wave turbulence still presents a number of theoretical challenges, one of them being the onset of turbulence. Wave turbulence occurs in systems of nonlinear dispersive waves, where energy transfer occurs chiefly among resonant sets of waves [Zakharov et al., 2004]. Wave turbulence is present in a plethora of physically relevant systems like capillary waves [Schröder et al., 1996], magnetized plasmas [Musher et al., 1995], superfluid helium [Kolmakov & Pokrowsky, 1995], nonlinear optics [Dyachenko et al., 1992], acoustic waves [Zakharov & Sagdeev, 1970], astrophysics [Sridhar & Goldreich, 1994], among others. In most applications of wave turbulence, the wave amplitudes are relatively weak, such that only quadratic nonlinearities need to be considered. Hence the dynamical features of more complicated models can be retained by simpler models, like the resonant three-wave interacting wave [Kaup et al., 1979]. Such system occurs in fluid dynamics [Turner, 1996; Li, 2007], plasma physics [Chian et al., 1994; Chian & Rizzato, 1994] and nonlinear optics [Rundquist et al., 1998; Stegeman & Segev, 1999; Picozzi & Haeltermann, 2001]. The nonlinear three-wave model describes the exchange of energy among a high-frequency (parent) wave and its sideband (daughters) with quadratic interactions, as well as with a spatial diffusion term. The model yet contains an energy source term which can be phenomenologically introduced from a linear growth rate for the parent wave. We identify the onset of wave turbulence as the excitation of spatial Fourier modes, in the presence of an underlying temporally chaotic dynamics. This approach can be pursued numerically by making a pseudo-spectral decomposition of the wave field. This procedure converts the nonlinear partial differential equation into a system of coupled nonlinear ordinary differential equations governing