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The Sagdeev-Zaslavski (SZ) equation for wave turbulence is analytically derived, both in terms of generating function and of multi-point pdf, for weakly interacting waves with initial random phases. When also initial amplitudes are random, the one-point pdf equation is derived. Such analytical calculations remarkably agree with results obtained in totally different fashions. Numerical investigations of the two-dimensional nonlinear Schroedinger equation (NLSE) and of a vibrating plate prove that: (i) generic Hamiltonian 4-wave systems rapidly attain a random distribution of phases independently of the slower dynamics of the amplitudes, vindicating the hypothesis of initially random phases; (ii) relaxation of the Fourier amplitudes to the predicted stationary distribution (exponential) happens on a faster timescale than relaxation of the spectrum (Rayleigh-Jeans distribution); (iii) the pdf equation correctly describes dynamics under different forcings: the NLSE has an exponential pdf corresponding to a quasi-gaussian solution, like the vibrating plates, that also show some intermittency at very strong forcings.
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The nonlinear dynamics of waves at the sea surface is believed to be ruled by the Weak Turbulence framework. In order to investigate the nonlinear coupling among gravity surface waves, we developed an experiment in the Coriolis facility which is a 13-m diameter circular tank. An isotropic and statistically stationary wave turbulence of average steepness of 10% is maintained by two wedge wave makers. The space and time resolved wave elevation is measured using a stereoscopic technique. Wave-wave interactions are analyzed through third and fourth order correlations. We investigate specifically the role of bound waves generated by non resonant 3-wave coupling. Specifically, we implement a space-time filter to separate the dynamics of free waves (i.e. following the dispersion relation) from the bound waves. We observe that the free wave dynamics causes weak resonant 4-wave correlations. A weak level of correlation is actually the basis of the Weak Turbulence Theory. Thus our observations support the use of the Weak Turbulence to model gravity wave turbulence as is currently been done in the operational models of wave forecasting. Although in the theory bound waves are not supposed to contribute to the energy cascade, our observation raises the question of the impact of bound waves on dissipation and thus on energy transfers as well.
A recent experiment [Deng et al., Nature 398, 218(1999)] demonstrated four-wave mixing of matter wavepackets created from a Bose-Einstein condensate. The experiment utilized light pulses to create two high-momentum wavepackets via Bragg diffraction from a stationary Bose-Einstein condensate. The high-momentum components and the initial low momentum condensate interact to form a new momentum component due to the nonlinear self-interaction of the bosonic atoms. We develop a three-dimensional quantum mechanical description, based on the slowly-varying-envelope approximation, for four-wave mixing in Bose-Einstein condensates using the time-dependent Gross-Pitaevskii equation. We apply this description to describe the experimental observations and to make predictions. We examine the role of phase-modulation, momentum and energy conservation (i.e., phase-matching), and particle number conservation in four-wave mixing of matter waves, and develop simple models for understanding our numerical results.
We consider a diatomic chain characterized by a cubic anharmonic potential. After diagonalizing the harmonic case, we study in the new canonical variables, the nonlinear interactions between the acoustical and optical branches of the dispersion relation. Using the {it wave turbulence} approach, we formally derive two coupled wave kinetic equations, each describing the evolution of the wave action spectral density associated to each branch. An $H$-theorem shows that there exist an irreversible transfer of energy that leads to an equilibrium solution characterized by the equipartition of energy in the new variables. While in the monoatomic cubic chain, in the large box limit, the main nonlinear transfer mechanism is based on four-wave resonant interactions, the diatomic one is ruled by a three wave resonant process (two acoustical and one optical wave): thermalization happens on shorter time scale for the diatomic chain with respect to the standard chain. Resonances are possible only if the ratio between the heavy and light masses is less than 3. Numerical simulations of the deterministic equations support our theoretical findings.
A fourth-order and a second-order nonlinear diffusion models in spectral space are proposed to describe gravitational wave turbulence in the approximation of strongly local interactions. We show analytically that the model equations satisfy the conservation of energy and wave action, and reproduce the power law solutions previously derived from the kinetic equations with a direct cascade of energy and an explosive inverse cascade of wave action. In the latter case, we show numerically by computing the second-order diffusion model that the non-stationary regime exhibits an anomalous scaling which is understood as a self-similar solution of the second kind with a front propagation following the law $k_f sim (t_*-t)^{3.296}$, with $t<t_*$. These results are relevant to better understand the dynamics of the primordial universe where potent sources of gravitational waves may produce space-time turbulence.
Via a sequence of approximations of the Lagrangian in Hamiltons principle for dispersive nonlinear gravity waves we derive a hierarchy of Hamiltonian models for describing wave-current interaction (WCI) in nonlinear dispersive wave dynamics on free surfaces. A subclass of these WCI Hamiltonians admits emph{emergent singular solutions} for certain initial conditions. These singular solutions are identified with a singular momentum map for left action of the diffeomorphisms on a semidirect-product Lie algebra. This semidirect-product Lie algebra comprises vector fields representing horizontal current velocity acting on scalar functions representing wave elevation. We use computational simulations to demonstrate the dynamical interactions of the emergent wavefront trains which are admitted by this special subclass of Hamiltonians for a variety of initial conditions. In particular, we investigate: (1) A variety of localised initial current configurations in still water whose subsequent propagation generates surface-elevation dynamics on an initially flat surface; and (2) The release of initially confined configurations of surface elevation in still water that generate dynamically interacting fronts of localised currents and wave trains. The results of these simulations show intricate wave-current interaction patterns whose structures are similar to those seen, for example, in Synthetic Aperture Radar (SAR) images taken from the space shuttle.
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