Interaction of a vortex ring with the free surface of ideal fluid

The interaction of a small vortex ring with the free surface of a perfect fluid is considered. In the frame of the point ring approximation the asymptotic expression for the Fourier-components of radiated surface waves is obtained in the case when the vortex ring comes from infinity and has both horizontal and vertical components of the velocity. The non-conservative corrections to the equations of motion of the ring, due to Cherenkov radiation, are derived.Comment: LaTeX, 15 pages, 1 eps figur

Dark matter-wave solitons in the dimensionality crossover

We consider the statics and dynamics of dark matter-wave solitons in the dimensionality crossover regime from 3D to 1D. There, using the nonpolynomial Schr\"{o}dinger mean-field model, we find that the anomalous mode of the Bogoliubov spectrum has an eigenfrequency which coincides with the soliton oscillation frequency obtained by the 3D Gross-Pitaevskii model. We show that substantial deviations (of order of 10% or more) from the characteristic frequency $\omega_{z}/\sqrt{2}$ ($\omega_{z}$ being the longitudinal trap frequency) are possible even in the purely 1D regime.Comment: Phys. Rev. A, in pres

Backlund transformations for the sl(2) Gaudin magnet

Elementary, one- and two-point, Backlund transformations are constructed for the generic case of the sl(2) Gaudin magnet. The spectrality property is used to construct these explicitly given, Poisson integrable maps which are time-discretizations of the continuous flows with any Hamiltonian from the spectral curve of the 2x2 Lax matrix.Comment: 17 pages, LaTeX, refs adde