Spectral bifurcations in dispersive wave turbulence

Dispersive wave turbulence is studied numerically for a class of one-dimensional nonlinear wave equations. Both deterministic and random (white noise in time) forcings are studied. Four distinct stable spectra are observed—the direct and inverse cascades of weak turbulence (WT) theory, thermal equilibrium, and a fourth spectrum (MMT; Majda, McLaughlin, Tabak). Each spectrum can describe long-time behavior, and each can be only metastable (with quite diverse lifetimes)—depending on details of nonlinearity, forcing, and dissipation. Cases of a long-live MMT transient state dcaying to a state with WT spectra, and vice-versa, are displayed. In the case of freely decaying turbulence, without forcing, both cascades of weak turbulence are observed. These WT states constitute the clearest and most striking numerical observations of WT spectra to date—over four decades of energy, and three decades of spatial, scales. Numerical experiments that study details of the composition, coexistence, and transition between spectra are then discussed, including: (i) for deterministic forcing, sharp distinctions between focusing and defocusing nonlinearities, including the role of long wavelength instabilities, localized coherent structures, and chaotic behavior; (ii) the role of energy growth in time to monitor the selection of MMT or WT spectra; (iii) a second manifestation of the MMT spectrum as it describes a self-similar evolution of the wave, without temporal averaging; (iv) coherent structures and the evolution of the direct and inverse cascades; and (v) nonlocality (in k-space) in the transferral process.

Brain-wave representation of words by superposition of a few sine waves

Data from three previous experiments were analyzed to test the hypothesis that brain waves of spoken or written words can be represented by the superposition of a few sine waves. First, we averaged the data over trials and a set of subjects, and, in one case, over experimental conditions as well. Next we applied a Fourier transform to the averaged data and selected those frequencies with high energy, in no case more than nine in number. The superpositions of these selected sine waves were taken as prototypes. The averaged unfiltered data were the test samples. The prototypes were used to classify the test samples according to a least-squares criterion of fit. The results were seven of seven correct classifications for the first experiment using only three frequencies, six of eight for the second experiment using nine frequencies, and eight of eight for the third experiment using five frequencies.