Dynamics of a homogeneous ball on a horizontal plane with sliding,spinning, and rolling friction taken into account

We analyze the dynamics of a homogeneous ball on a horizontal plane with friction of all kinds, namely, sliding, spinning, and rolling friction, taken into account. The qualitative-analytic study of the ball dynamics is supplemented with numerical experiments.     

The interconnection of sliding and rolling in the problem of the motion of a homogeneous sphere on a rough horizontal plane

The motion of a homogeneous sphere on a rough horizontal plane when the angular velocities of the twisting and spinning of the sphere are equal to zero at the initial instant is considered. It is proved that, for any initial conditions, the angular velocity of the rolling of the sphere and the sliding velocity vanish after the same finite time. It is shown that the sliding and rolling are interconnected and, in particular, that the rolling of a sphere without sliding is impossible.     

Rosen-Zener model in cold molecule formation

The Rosen-Zener model for association of atoms in a Bose-Einstein condensate is studied. Using a nonlinear Volterra integral equation, we obtain an analytic formula for final probability of the transition to the molecular state for weak interaction limit. Considering the strong coupling limit of high field intensities, we show that the system reveals two different time-evolution pictures depending on the detuning of the frequency of the associating field. For both limit cases we derive highly accurate formulas for the molecular state probability valid for the whole range of variation of time. Using these formulas, we show that at large detuning regime the molecule formation process occurs almost non-oscillatory in time and a Rosen-Zener pulse is not able to associate more than one third of atoms at any time point. The system returns to its initial all-atomic state at the end of the process and the maximal transition probability 1/6 is achieved when the field intensity reaches its peak. In contrast, at small detuning the evolution of the system displays large-amplitude oscillations between atomic and molecular populations. We find that the shape of the oscillations in the first approximation is defined by the field detuning only. Finally, a hidden singularity of the Rosen-Zener model due to the specific time-variation of the field amplitude at the beginning of the interaction is indicated. It is this singularity that stands for many of the qualitative and quantitative properties of the model. The singularity may be viewed as an effective resonance-touching.     

Second-harmonic generation in a layer with variable susceptibility

The weak nonlinearity limit of the second-harmonic generation in a medium with variable second-order susceptibility, changing along the direction of propagation of interacting waves according to the hyperbolic secant law, is considered. It is shown that the variation of the second harmonic’s normalized intensity with the spatial coordinate is given in terms of the solution to an auxiliary linear problem, which is known as the Rosen-Zener quantum two-level model. The final intensity of the second harmonic at the exit from the medium is calculated and analyzed. It is shown that, for the particular Rosen-Zener profile under consideration, because of its inherent properties, the variation range of the peak susceptibility of the layer for which the medium can be strictly considered as weakly nonlinear is narrower than the corresponding parameter range for a homogeneous medium.     

Dynamic chaos in the solution of the Gross-Pitaevskii equation for a periodic potential

We analytically and numerically investigate the solution to the stationary Gross-Pitaevskii equation for a one-dimensional potential of the optical lattice in the case of repulsive nonlinearity. From the mathematical viewpoint, this problem is similar to the well-known problem of the classical mathematical Kapitza pendulum perturbed by a weak high-frequency force. At certain values of the parameters, dynamic chaos is produced in the considered problem. It is modeled analytically by a nonlinear diffusion equation.     

Qualitative Consideration of the Scharnhorst Effect

JETP Letters - The Scharnhorst effect—a photon flying across a gap between large conducting plates parallel to each other has a speed slightly higher than the speed of light in vacuum because...     

Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions

Considering the equations for some functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented.     

Two-state model of a general Heun class with periodic level-crossings

A specific constant-amplitude periodic level-crossing model of the semi-classical quantum time-dependent two-state problem that belongs to a general Heun class of field configurations is presented. The exact analytic solution for the probability amplitude, generally written for this class in terms of the general Heun functions, in this specific case admits series expansion in terms of the incomplete Beta functions. Terminating this series results in an infinite hierarchy of finite-sum closedform solutions each standing for a particular two-state model, which generally is only conditionally integrable in the sense that for these field configurations the amplitude and phase modulation functions are not varied independently. However, there exists at least one exception when the model is unconditionally integrable, that is the Rabi frequency and the detuning of the driving optical field are controlled independently. This is a constant-amplitude periodic level-crossing model, for which the detuning in a limit becomes a Dirac delta-comb configuration with variable frequency of the levelcrossings. The exact solution for this model is derived, the Floquet exponents are determined and study of the population dynamics in the system for various regions of the input parameters is done.     

Dissociation of Quarkonium in a Strong Electric Field

The probability of tunnel decay of quarkonium (bond state of a heavy quark and a heavy antiquark) into free quarks in a strong electric field is estimated.