Evolution of non-stationary pulses in a cold magnetized quark-gluon plasma

We study weakly nonlinear wave perturbations propagating in a cold nonrelativistic and magnetized ideal quark-gluon plasma. We show that such perturbations can be described by the Ostrovsky equation. The derivation of this equation is presented for the baryon density perturbations. Then we show that the generalized nonlinear Schr{\"o}dinger (NLS) equation can be derived from the Ostrovsky equation for the description of quasi-harmonic wave trains. This equation is modulationally stable for the wave number $k < k_m$ and unstable for $k > k_m$, where $k_m$ is the wave number where the group velocity has a maximum. We study numerically the dynamics of initial wave packets with the different carrier wave numbers and demonstrate that depending on the initial parameters they can evolve either into the NLS envelope solitons or into dispersive wave trains

Soliton spectra of random water waves in shallow basins

Interpretation of random wave field on a shallow water in terms of Fourier spectra is not adequate, when wave amplitudes are not infinitesimally small. A nonlinearity of wave fields leads to the harmonic interactions and random variation of Fourier spectra. As has been shown by Osborne and his co-authors, a more adequate analysis can be performed in terms of nonlinear modes representing cnoidal waves; a spectrum of such modes remains unchanged even in the process of nonlinear mode interactions. Here we show that there is an alternative and more simple analysis of random wave fields on shallow water, which can be presented in terms of interacting Korteweg - de Vries solitons. The data processing of random wave field is developed on the basis of inverse scattering method. The soliton component obscured in a random wave field is determined and a corresponding distribution function of number of solitons on their amplitudes is constructed. The approach developed is illustrated by means of artificially generated quasi-random wave field and applied to the real data interpretation of wind waves generated in the laboratory wind tank.Comment: 23 pages, 15 figure

Description of vortical flows of incompressible fluid in terms of quasi-potential function

It has been shown [1, 2] that a wide class of 3D motions of in- compressible viscous fluid in Cartesian coordinates can be de- scribed by only one scalar function dubbed the quasi-potential. This class of fluid flows is characterized by three-component velocity field having two-component vorticity field; both these fields can depend of all three spatial variables and time, in gen- eral. Governing equations for the quasi-potential have been de- rived and simple illustrative examples of 3D flows have been presented. In this paper the concept of quasi-potential is fur- ther developed for fluid flows in cylindrical coordinates. It is shown that the introduction of a quasi-potential in curvilinear coordinates is non-trivial and may be a subject of additional restrictions. In the cases when it is possible, we construct il- lustrative examples which can be of interest for some practical applications

The inverse problem for the Gross - Pitaevskii equation

Two different methods are proposed for the generation of wide classes of exact solutions to the stationary Gross - Pitaevskii equation (GPE). The first method, suggested by the work by Kondrat'ev and Miller (1966), applies to one-dimensional (1D) GPE. It is based on the similarity between the GPE and the integrable Gardner equation, all solutions of the latter equation (both stationary and nonstationary ones) generating exact solutions to the GPE, with the potential function proportional to the corresponding solutions. The second method is based on the "inverse problem" for the GPE, i.e. construction of a potential function which provides a desirable solution to the equation. Systematic results are presented for 1D and 2D cases. Both methods are illustrated by a variety of localized solutions, including solitary vortices, for both attractive and repulsive nonlinearity in the GPE. The stability of the 1D solutions is tested by direct simulations of the time-dependent GPE

Interaction of Kortewegâ-de Vries solitons with external sources

We consider the problem of interaction of a solitary wave with a moving external source within the framework of Korteweg– de Vries (KdV) equation. We show that for certain profiles of external source the problem has exact solutions in the form of a stationary solitary waves coupled with the force. For the solitary waves which are not trapped by the external force of a small amplitude we obtain approximate solutions by means of the asymptotic method and analyse solutions with the arbi- trary relationship between the widths of forcing function and solitary wave. Results obtained agree well with the results of previous works where only the limiting cases of very narrow or infinitely wide forcing as compared with the width of soli- tary wave were studied. Several new regimes of soliton interac- tion with width the forcing have been revealed. The theoretical results have been validated by the direct numerical modelling within the framework of forced KdV equation

Dynamics of two charged particles in a creeping flow

We study the interaction of two charged solid particles in a viscous fluid. It is assumed that the particles move either side-by-side or one after another along the same vertical line under the influence of the buoyancy/gravity force, Coulomb electrostatic force or its modification, and viscous drag force. The drag force consists of two components: the quasi-stationary Stokes drag force and Boussinesq-Basset drag force resulting from the unsteady motion. Solutions of the governing equations are analysed analytically and numerically for the cases of perfect fluid and viscous fluid; the comparison of these two cases is presented

Internal solitary waves in two-layer fluids at near-critical situation

A new model equation describing weakly nonlinear long internal waves at the interface between two thin layers of different density is derived for the specific relationships between the densities, layer thicknesses and surface tension between the layers. The equation derived and dubbed here the Gardner–Kawahara equation represents a natural generalisation of the well-known Korteweg–de Vries (KdV) equation containing the cubic nonlinear term as well as fifth-order dispersion term. Solitary wave solutions are investigated numerically and categorised in terms of two dimensionless parameters, the wave speed and fifth-order dispersion. The equation derived may be applicable to wave description in other media