Dyadic bilinear estimates and applications to the well-posedness for the 2D Zakharov–Kuznetsov equation in the endpoint space 𝐻−1/4

The Cauchy problem of the 2D Zakharov–Kuznetsov equation {\partial_{t}u+\partial_{x}(\partial_{xx}+\partial_{yy})u+uu_{x}=0} is considered. It is shown that the 2D Z-K equation is locally well-posed in the endpoint Sobolev space {H^{-1/4}}, and it is globally well-posed in {H^{-1/4}} with small initial data. In this paper, we mainly establish some new dyadic bilinear estimates to obtain the results, where the main novelty is to parametrize the singularity of the resonance function in terms of a univariate polynomial.

Ambiguity in Tempo Perception: What Draws Listeners to Different Metrical Levels?

The distribution of listeners’ perceived tempi across large collections of music has been modeled previously by a resonance function with a peak near the “preferred tempo” of 120 beats per minute (BPM) [Van Noorden and Moelants, J. New Music Res., 28, 43–66]. Here, through a series of experiments in which listeners were asked to tap to the most salient pulse of musical excerpts,we examined distributions of tapped tempi from single musical excerpts to see if the global resonance of preferred tempo is dependent on musical content. Results show that for some musical excerpts, the distribution of perceived tempi conforms to the global resonant form in that metrical levels with tempi near 120 BPM were perceived as most salient, while for other excerpts the most saliently perceived tempo sat well above or below 120 BPM. We then used a model, which quantifies relative strengths of periodicities in the audio signal, to demonstrate that deviations from the “preferred tempo” can be partially explained by dynamic rhythmic accents drawing listeners to tempi away from the resonance.

On Dupree's resonance function

It is shown that Dupree's resonance function has a negative real asymptotic tail, so that the dispersion relation of the renormalized weak turbulence theory leads to unstable high phase velocity waves, even when the average distribution is a Gaussian. A possible explanation of this paradox is proposed.

Nonlinear theory of a whistler wave

Using the Dupree—Weinstock perturbed-orbit model of plasma turbulence, we obtain the diffusion equation describing the evolution of the average one-particle distribution function for whistler mode turbulence. The numerical result for electron pitch-angle diffusion within this scheme leads us to conclude that the effect of the resonance broadening due to perturbed orbits on the pitch-angle diffusion coefficient is not large compared with that evaluated by the unperturbed orbit in the whistler mode spectrum with a finite width. Based on the explicitly evaluated resonance function, the effects of this broadening on the growth rate for the whistler wave are also discussed.

The two stream instability in the ‘resonance’ model

In this note the problem of stability of two hot collisionless streams of charged particles is considered. The masses, charges, densities, and temperatures are arbitrary and the distribution functions are modelled by one ‘resonance’ function for each stream. The problem of stability is resolved by Nyquist diagrams, and, for the case of equal plasma frequencies, also by solving the dispersion relation in ω. A comparison with two Maxwellians on the one hand, and a two step function model on the other, is given. Step functions appear to be too crude for this problem.