Euler equations on homogeneous spaces and Virasoro orbits

We show that the following three systems related to various hydrodynamical approximations: the Korteweg--de Vries equation, the Camassa--Holm equation, and the Hunter--Saxton equation, have the same symmetry group and similar bihamiltonian structures. It turns out that their configuration space is the Virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right-invariant metrics on this group or on appropriate homogeneous spaces. In particular, we describe how Arnold's approach to the Euler equations as geodesic flows of one-sided invariant metrics extends from Lie groups to homogeneous spaces. We also show that the above three cases describe all generic bihamiltonian systems which are related to the Virasoro group and can be integrated by the translation argument principle: they correspond precisely to the three different types of generic Virasoro orbits.Comment: 26 pages, 4 figures, LaTeX. Advances in Mathematics (to appear

Geodesic Flow on the Diffeomorphism Group of the circle

We show that certain right-invariant metrics endow the infinite-dimensional Lie group of all smooth orientation-preserving diffeomorphisms of the circle with a Riemannian structure. The study of the Riemannian exponential map allows us to prove infinite-dimensional counterparts of results from classical Riemannian geometry: the Riemannian exponential map is a smooth local diffeomorphism and the length-minimizing property of the geodesics holds.Comment: 15 page

An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion

We study a class of 1+1 quadratically nonlinear water wave equations that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation, yet still preserves integrability via the inverse scattering transform (IST) method. This IST-integrable class of equations contains both the KdV equation and the CH equation as limiting cases. It arises as the compatibility condition for a second order isospectral eigenvalue problem and a first order equation for the evolution of its eigenfunctions. This integrable equation is shown to be a shallow water wave equation derived by asymptotic expansion at one order higher approximation than KdV. We compare its traveling wave solutions to KdV solitons.Comment: 4 pages, no figure

Peakons arising as particle paths beneath small-amplitude water waves

We present a new kind of particle path in constant vorticity water of finite depth, within the framework of small-amplitude waves

Influence of aging on medial olivocochlear system function

Grazyna Lisowska, Grzegorz Namyslowski, Boguslawa Orecka, Maciej MisiolekDepartment of Otolaryngology, Medical University of Silesia, Zabrze, PolandBackground: There is still controversy regarding the influence of aging on medial olivocochlear (MOC) system function. The main objective of this study is to measure age-related changes of MOC system function in people with normal hearing thresholds.Method: Bilateral assessment of the MOC effect for click-evoked otoacoustic emissions (CEOAEs; at 70±3 dB peak sound pressure level [pSPL], click at 50/second, 260 repeats, 2.5–20 millisecond window) and for distortion product otoacoustic emissions (DPOAEs; with [frequencies] f2/f1=1.22, [levels of primary tones] L1=65 dB SPL and L2=55 dB SPL; DP-grams for 2f1-f2 were collected for the f1 frequencies varying from 977 Hz to 5,164 kHz, with the resolution of four points per octave) was performed in a group of 146 (n=292 ears) healthy, right-handed subjects aged from 10–60 years with a bilateral hearing threshold from 0.25–4.0 kHz, not exceeding 20 dB hearing level; normal tympanograms; and a threshold of the contralateral stapedial reflex for broadband noise (BBN) of 75 dB SPL or higher. The MOC inhibition was assessed on the basis of changes in OAE level during BBN contralateral stimulation at 50 dB sensation level (mean, 65±3 dB SPL).Results: Comparative analysis of the MOC effect for CEOAE and DPOAE showed the weakest effect in the oldest age group (41–60 years) at almost all tested frequencies. Moreover, a weak, albeit significant, positive correlation between the level of OAE and the size of the MOC effect was documented.Conclusion: On the basis of our study, we have found a decrease in the strength of the MOC system with increasing age in normally hearing subjects, as reflected by a decrease of the OAE suppression effects in older individuals and an increase of the number of CEOAE and DPOAE enhancements during contralateral acoustic stimulation in the elderly, especially in the high-frequency range.Keywords: contralateral suppression, efferent suppression, MOC effect, MOC inhibition, otoacoustic emissions, medial olivocochlear system, ag