Matter Wave Solitons at Finite Temperatures

We consider the dynamics of a dark soliton in an elongated harmonically trapped Bose-Einstein condensate. A central question concerns the behavior at finite temperatures, where dissipation arises due to the presence of a thermal cloud. We study this problem using coupled Gross-Pitaevskii and N-body simulations, which include the mean field coupling between the condensate and thermal cloud. We find that the soliton decays relatively quickly even at very low temperatures, with the decay rate increasing with rising temperature.     

Collision dynamics of elliptically polarized solitons in Coupled Nonlinear Schrödinger Equations

We investigate numerically the collision dynamics of elliptically polarized solitons of the System of Coupled Nonlinear Schrödinger Equations (SCNLSE) for various different initial polarizations and phases. General initial elliptic polarizations (not sech$sech$-shape) include as particular cases the circular and linear polarizations. The elliptically polarized solitons are computed by a separate numerical algorithm. We find that, depending on the initial phases of the solitons, the polarizations of the system of solitons after the collision change, even for trivial cross-modulation. This sets the limits of practical validity of the celebrated Manakov solution. For general nontrivial cross-modulation, a jump in the polarization angles of the solitons takes place after the collision (‘polarization shock’). We study in detail the effect of the initial phases of the solitons and uncover different scenarios of the quasi-particle behavior of the solution. In majority of cases the solitons survive the interaction preserving approximately their phase speeds and the main effect is the change of polarization. However, in some intervals for the initial phase difference, the interaction is ostensibly inelastic: either one of the solitons virtually disappears, or additional solitons are born after the interaction. This outlines the role of the phase, which has not been extensively investigated in the literature until now.     

A note on the Painlevé test for nonlinear variable-coefficient PDEs

For a generalized nonlinear PDEs with variable coefficients, it is not Painlevé integrable unless the variable coefficients satisfy certain constraint conditions. In this note a generalized algorithm is proposed for the Painlevé test of nonlinear variable-coefficient PDEs. For the three steps of Painlevé test, i.e. leading order analysis, resonance determination and verification of resonant conditions, the analysis of parametric constraints is similar to those of nonlinear PDEs with constant coefficients given in my previous work. The main difference lies in the coefficients of Laurent series should have proper dependence according to the types of variable coefficients. By this generalized algorithm, several important nonlinear variable-coefficient PDEs, including KdV equation, mKdV equation, KP equation, NLS equation and higher-order NLS equation are studied and, in addition to rederiving all known P-integrable conditions, some new P-integrable models are obtained with the assistance of Maple.     

Multisymplectic numerical method for the regularized long-wave equation

In this paper, we derive a 6-point multisymplectic Preissman scheme for the regularized long-wave equation from its Bridges' multisymplectic form. Backward error analysis is implemented for the new scheme. The performance and the efficiency of the new scheme are illustrated by solving several test examples. The obtained results are presented and compared with previous methods. Numerical results indicate that the new multisymplectic scheme can not only obtain satisfied solutions, but also keep three invariants of motion very well.     

Scattering of Cold Neutrons on Gel Samples Formed by Impurity Clusters in Superfluid He-II

The approach we have proposed earlier for a large increase in ultracold neutron density was based on equilibrium thermalization of cold neutrons during their diffusive motion in a large sample of an impurity-helium gel which consists of weakly bounded deuterium or heavy water nanoclusters in He-II cooled below a few mK. It is clear that the nanoparticles should provide a sufficiently large cross-section of coherent scattering of cold neutrons, but the scattering of neutrons on impurity gel samples in He-II has been never investigated earlier. In this paper we present results of the first observations of the scattering of a beam of cold neutrons with mean velocities from 30 to 160 m/s on heavy water and deuterium gel samples in He-II at temperatures T∼1.6 K. It has been observed that the beam transmission through the heavy water gel sample has been changed from ∼0% to ∼50% upon increasing the neutron velocity from 30 to 160 m/s, and that the neutron transmission trough the deuterium gel sample is less than 5% at all velocities. The angular distribution of the neutron scattering changes with variations in the neutron velocity: for both the gels at large neutron energies the scattering is directed in a front hemisphere mainly, and at low energies the angular distribution of scattered neutrons is close to uniform.     

Recent Progress in Studies of Nanostructured Impurity–Helium Solids

Impurity–helium (Im–He) solids are porous materials formed inside superfluid ⁴He by nanoclusters of impurities injected from the gas phase. The results of studies of these materials have relevance to soft condensed matter physics, matrix isolation of free radicals and low temperature chemistry. Recent studies by a variety of experimental techniques, including CW and pulse ESR, X-ray diffraction, ultrasound and Raman spectroscopy allow a better characterization of the properties of Im–He solids. The structure of Im–He solids, the trapping sites of stabilized atoms and the possible energy content of the samples are analyzed on the basis of experimental data. The kinetics of exchange tunneling reactions of hydrogen isotopes in nanoclusters and the changes of environment of the atoms during the course of these reactions are reviewed. Analysis of the ESR data shows that very large fraction of the stabilized atoms in Im–He solids reside on the surfaces of impurity nanoclusters. The future directions for studying Im–He solids are described. Among the most attractive are the studies of Im–He solids with high concentrations of stabilized atoms at ultralow (10–20 mK) temperature for the observation of new collective quantum phenomena, the studies of practical application of Im–He solids as a medium in neutron moderator for efficient production of ultracold (∼1 mK) neutrons, and the possibilities of obtaining high concentration of atomic nitrogen embedded in N₂ clusters for energy storage.     

Numerical realizations of solutions of the stochastic KdV equation

We investigate simulations of exact solutions of the stochastic Korteweg–deVries equation under additive noise. We compare the expectation values of the exact solutions to theoretical expectation values and to the numerical simulations of the stochastic Korteweg–deVries equation with and without damping. We find on average the diffused soliton vanishes long before the typically reported asymptotic limit.     

Hidden solitons in the Zabusky–Kruskal experiment: Analysis using the periodic,inverse scattering transform

Recent numerical work on the Zabusky–Kruskal experiment has revealed, amongst other things, the existence of hidden solitons in the wave profile. Here, using Osborne’s nonlinear Fourier analysis, which is based on the periodic, inverse scattering transform, the hidden soliton hypothesis is corroborated, and the exact number of solitons, their amplitudes and their reference level is computed. Other “less nonlinear” oscillation modes, which are not solitons, are also found to have nontrivial energy contributions over certain ranges of the dispersion parameter. In addition, the reference level is found to be a non-monotone function of the dispersion parameter. Finally, in the case of large dispersion, we show that the one-term nonlinear Fourier series yields a very accurate approximate solution in terms of Jacobian elliptic functions.