A characterization of isometries of CAT(0)-space as maps preserving diagonal tube

We give positive answers for questions by Berestovskii. Namely, we prove that every bijection of locally compact geodesically complete and connected at infinity CAT(0)-space $X$ onto itself preserving some fixed distance or satellite relations is an isometry of this space. The proof of this theorem is based on another result stated by Berestovskii as a problem: the metric of the space $X$ may be recovered from its diagonal tube corresponding to an arbitrary number $r > 0$.Comment: 28 page

Extraordinary SEAWs under influence of the spin-spin interaction and the quantum Bohm potential

The separate spin evolution (SSE) of electrons causes the existence of the spin-electron acoustic wave. Extraordinary spin-electron acoustic waves (SEAWs) propagating perpendicular to the external magnetic field have large contribution of the transverse electric field. Its spectrum has been studied in the quasi-classical limit at the consideration of the separate spin evolution. The spin-spin interaction and the quantum Bohm potential give contribution in the spectrum extraordinary SEAW. This contribution is studied in this paper. Moreover, it is demonstrated that the spin-spin interaction leads to the existence of the extraordinary SEAWs if the SSE is neglected. The hybridization of the extraordinary SEAW and the lower extraordinary wave in the regime, where the cyclotron frequency is larger then the Langmuir frequency is studied either.Comment: 8 pages, 8 figure

Spin current contribution in the spectrum of collective excitations of degenerate partially polarized spin-1/2 fermions at separate dynamics of spin-up and spin-down fermions

The spectrum of collective excitations of degenerate partially polarized spin-1/2 fermions is considered. The spin-up fermions and the spin-down fermions are considered as different fluids. Corresponding two-fluid hydrodynamics consistent with a non-linear Pauli equation is suggested. An equation of state for the spin current caused by the distribution of particles on different energy levels is suggested for the degenerate regime, where the spin current is caused by the Pauli blocking. Spectrum of three waves is found as a solution of the hydrodynamic equations: two sound waves and one spin wave. Their spectrums are calculated for two regimes: propagation parallel and perpendicular to the direction of the equilibrium spin polarization.Comment: 14 pages, 15 figure

Quantum hydrodynamic theory of quantum fluctuations in dipolar Bose-Einstein condensate

Traditional quantum hydrodynamics of Bose-Einstein condensates (BECs) is restricted by the continuity and Euler equations. It corresponds to the well-known Gross-Pitaevskii equation. However, the quantum Bohm potential, which is a part of the momentum flux, has a nontrivial part with can evolve under the quantum fluctuations. To cover this phenomenon in terms of hydrodynamic methods we need to derive equations for the momentum flux, and the third rank tensor. In all equations we consider the main contribution of the short-range interaction (SRI) in the first order by the interaction radius. Derived hydrodynamics consists of four hydrodynamic equations. The third moment evolution equation contains interaction leading to the quantum fluctuations. It includes new interaction constant. The Gross-Pitaevskii interaction constant is the integral of potential, but the second interaction constant is the integral of second derivative of potential. If we have dipolar BECs we deal with a long-range interaction. Its contribution is proportional to the potential of dipole-dipole interaction (DDI). The Euler equation contains the derivative of the potential. The third rank tensor evolution equation contains the third derivative of the potential. The quantum fluctuations lead to existence of the second wave solution. Moreover, the quantum fluctuations introduce the instability of BECs. If the DDI is attractive, but being smaller then the repulsive SRI presented by the first interaction constant, there is the long-wavelength instability. For the repulsive DDI these is more complex picture. There is the small area with the long-wavelength instability which transits into stability interval, where two waves exist. There is the short-wavelength instability as well. These results are found for the DDI strength comparable with the Gross-Pitaevskii SRI.Comment: 6+3 pages, 3+2 figure

Spin electron acoustic soliton: Separate spin evolution of electrons with exchange interaction

Separate spin evolution quantum hydrodynamics is generalized to include the Coulomb exchange interaction. The Coulomb exchange interaction is considered as the interaction between the spin-down electrons being in the quantum states occupied by one electron, giving main contribution in the equilibrium. The generalized model is applied to study the non-linear spin-electron acoustic waves. Existence of the spin-electron acoustic soliton is demonstrated. Contributions of the concentration, spin polarization, and exchange interaction in the properties of the spin electron acoustic soliton are studied.Comment: 10 pages, 6 figure

Kinetic description of the oblique propagating spin-electron acoustic waves in degenerate plasmas

Oblique propagation of the spin-electron acoustic waves in degenerate magnetized plasmas is considered in terms of quantum kinetics with the separate spin evolution, where the spin-up electrons and the spin-down electrons are considered as two different species with different equilibrium distributions. It is considered in the electrostatic limit. Corresponding dispersion equation is derived. Analytical analysis of the dispersion equation is performed in the long-wavelength limit to find an approximate dispersion equation describing the spin-electron acoustic wave. The approximate dispersion equation is solved numerically. Real and imaginary parts of the spin-electron acoustic wave frequency are calculated for different values of the parameters describing the system. It is found that the increase of angle between direction of wave propagation and the external magnetic field reduces the real and imaginary parts of spin-electron acoustic wave frequency. The increase of the spin polarization decreases the real and imaginary parts of frequency either. The imaginary part of frequency has nonmonotonic dependence on the wave vector which shows a single maximum. The imaginary part of frequency is small in compare with the real part for all parameters in the area of applicability of the obtained dispersion equation.Comment: 8 pages, 7 figure

Non-integral form of the Gross-Pitaevskii equation for polarized molecules

The Gross-Pitaevskii equation for polarized molecules is an integro-differential equation, consequently it is complicated for solving. We find a possibility to represent it as a non-integral nonlinear Schrodinger equation, but this equation should be coupled with two linear equations describing electric field. These two equations are the Maxwell equations. We recapture the dispersion of collective excitations in the three dimensional electrically polarized BEC with no evolution of the electric dipole moment directions. We trace the contribution of the electric dipole moment. We explicitly consider the contribution of the electric dipole moment in the interaction constant for the short-range interaction. We show that the spectrum of dipolar BEC reveals no instability at repulsive short-range interaction. Nonlinear excitations are also considered. We present dependence of the bright soliton characteristics on the electric dipole moment.Comment: 7 pages. arXiv admin note: text overlap with arXiv:1107.202

Hydrodynamic and kinetic models for spin-1/2 electron-positron quantum plasmas: Annihilation interaction, helicity conservation, and wave dispersion in magnetized plasmas

We discuss complete theory of spin-1/2 electron-positron quantum plasmas, when electrons and positrons move with velocities mach smaller than the speed of light. We derive a set of two fluid quantum hydrodynamic equations consisting of the continuity, Euler, spin (magnetic moment) evolution equations for each species. We explicitly include the Coulomb, spin-spin, Darwin and annihilation interactions. The annihilation interaction is the main topic of the paper. We consider contribution of the annihilation interaction in the quantum hydrodynamic equations and in spectrum of waves in magnetized electron-positron plasmas. We consider propagation of waves parallel and perpendicular to an external magnetic field. We also consider oblique propagation of longitudinal waves. We derive set of quantum kinetic equations for electron-positron plasmas with the Darwin and annihilation interactions. We apply the kinetic theory for the linear wave behavior in absence of external fields. We calculate contribution of the Darwin and annihilation interactions in the Landau damping of the Langmuir waves. We should mention that the annihilation interaction does not change number of particles in the system. It does not related to annihilation itself, but it exists as a result of interaction of an electron-positron pair via conversion of the pair into virtual photon. A pair of the non-linear Schrodinger equations for electron-positron plasmas including the Darwin and annihilation interactions. Existence of conserving helicity in electron-positron quantum plasmas of spinning particles with the Darwin and annihilation interactions is demonstrated. We show that annihilation interaction plays an important role in quantum electron-positron plasmas giving contribution of the same magnitude as the spin-spin interaction.Comment: 21 pages, 13 figure

First principles derivation of NLS equation for BEC with cubic and quintic nonlinearities at non zero temperature. Dispersion of linear waves

In this work we presented a derivation of the quantum hydrodynamic equations for neutral bosons. We considered short range interaction between particles. This interaction consist binary interaction $U(\textbf{r}_{i},\textbf{r}_{j})$ and three particle interaction $U(\textbf{r}_{i},\textbf{r}_{j},\textbf{r}_{k})$, the last one does not include binary interaction between particles. From the quantum hydrodynamic (QHD) equations for Bose-Einstein condensate we derive nonlinear Schr\"{o}dinger equation. This equation includes the nonlinearities of third and fifth degree. It is at zero temperature. Explicit form of the constant of three-particle interaction was taken. First of all, developed method we used for studying of dispersion of linear waves. Dispersion characteristics of linear waves were compared for the cases. It were of two-particle interaction in approximation third order to interaction radius (TOIR) and three-particle interaction, at zero temperature. We consider influence of temperature on dispersion of elementary excitations. For this aim we derive a system of QHD equations at non-zero temperature. Obtained system of equation is an analog of well-known two-fluid hydrodynamics. Moreover, it is generalization of two-fluid hydrodynamics equations due to three-particle interaction. Evident expressions of the velocities of the first and second sound via the concentrations of superfluid and noncondesate components is calculated.Comment: 14 page

The quantum hydrodynamic description of quantum gases with different interactions

We describe recent development of quantum hydrodynamics for ultracold Bose particle studying and consider different kinds of interactions. The method of derivation of equations describing the evolution of the neutral Bose particle system at low temperatures is described. Despite the fact that we consider the neutral particles we account the short-range interaction between particles. We consider the particles in the Bose-Einstein condensate (BEC) state. This method is called the method of quantum hydrodynamics, because natural for of the quantum mechanics rewritten in terms of material fields of observable quantities in three dimensional space is the set of equations, which look like the hydrodynamics equations. It can be shown that from the quantum hydrodynamics equations can be derived macroscopic non-linear Schrodinger equation. Most famous non-linear Schrodinger equation is the Gross-Pitaevskii (GP) equation, which contains nonlinearity of the third degree. There are generalizations of the GP equation. New term appears in the GP equation at account of the three-particle interaction. This term contains nonlinearity of the fifth degree. At more detailed account of the two particle interaction we come to the nonlocal non-linear Schrodinger equation. This equation contains spatial derivatives of the order parameter in the non-linear terms caused by the interaction. In this terminology the GP equation corresponds to the first order by the interaction radius. For the BEC of the neutral particles with anisotropic long-range dipole-dipole interaction the generalization of the GP equation was also suggested. Detailed analyses of the applicability conditions shows that this equation valid for the system of dipoles parallel to each other, which do not change their direction, and where the dipole-dipole interaction interferences translational motion of particles.Comment: 37 page