On the classification of discrete Hirota-type equations in 3D

In the series of recent publications we have proposed a novel approach to the classification of integrable differential/difference equations in 3D based on the requirement that hydrodynamic reductions of the corresponding dispersionless limits are `inherited' by the dispersive equations. In this paper we extend this to the fully discrete case. Our only constraint is that the initial ansatz possesses a non-degenerate dispersionless limit (this is the case for all known Hirota-type equations). Based on the method of deformations of hydrodynamic reductions, we classify discrete 3D integrable Hirota-type equations within various particularly interesting subclasses. Our method can be viewed as an alternative to the conventional multi-dimensional consistency approach.Comment: 29 page

Transport of the repulsive Bose-Einstein condensate in a double-well trap: interaction impact and relation to Josephson effect

Two aspects of the transport of the repulsive Bose-Einstein condensate (BEC) in a double-well trap are inspected: impact of the interatomic interaction and analogy to the Josephson effect. The analysis employs a numerical solution of 3D time-dependent Gross-Pitaevskii equation for a total order parameter covering all the trap. The population transfer is driven by a time-dependent shift of a barrier separating the left and right wells. Sharp and soft profiles of the barrier velocity are tested. Evolution of the relevant characteristics, involving phase differences and currents, is inspected. It is shown that the repulsive interaction substantially supports the transfer making it possible i) in a wide velocity interval and ii) three orders of magnitude faster than in the ideal BEC. The transport can be approximately treated as the d.c. Josephson effect. A dual origin of the critical barrier velocity (break of adiabatic following and d.c.-a.c. transition) is discussed. Following the calculations, robustness of the transport (d.c.) crucially depends on the interaction and barrier velocity profile. Only soft profiles which minimize undesirable dipole oscillations are acceptable.Comment: 10 pages, 8 figures, accepted by Laser Physis. arXiv admin note: text overlap with arXiv:1312.2750 The replaced version has a few corrections and additional reference

Once more on extra quark-lepton generations and precision measurements

Precision measurements of $Z$-boson parameters and $W$-boson and $t$-quark masses put strong constraints on non $SU(2)\times U(1)$ singlet New Physics. We demonstrate that one extra generation passes electroweak constraints even when all new particle masses are well above their direct mass bounds.Comment: Dedicated to L.B. Okun's 80th birthda

On the classification of scalar evolutionary integrable equations in 2+12+1 dimensions

We consider evolutionary equations of the form $u_t=F(u, w)$ where $w=D_x^{-1}D_yu$ is the nonlocality, and the right hand side $F$ is polynomial in the derivatives of $u$ and $w$. The recent paper \cite{FMN} provides a complete list of integrable third order equations of this kind. Here we extend the classification to fifth order equations. Besides the known examples of Kadomtsev-Petviashvili (KP), Veselov-Novikov (VN) and Harry Dym (HD) equations, as well as fifth order analogues and modifications thereof, our list contains a number of equations which are apparently new. We conjecture that our examples exhaust the list of scalar polynomial integrable equations with the nonlocality $w$. The classification procedure consists of two steps. First, we classify quasilinear systems which may (potentially) occur as dispersionless limits of integrable scalar evolutionary equations. After that we reconstruct dispersive terms based on the requirement of the inheritance of hydrodynamic reductions of the dispersionless limit by the full dispersive equation

Anomalous scaling in two and three dimensions for a passive vector field advected by a turbulent flow

A model of the passive vector field advected by the uncorrelated in time Gaussian velocity with power-like covariance is studied by means of the renormalization group and the operator product expansion. The structure functions of the admixture demonstrate essential power-like dependence on the external scale in the inertial range (the case of an anomalous scaling). The method of finding of independent tensor invariants in the cases of two and three dimensions is proposed to eliminate linear dependencies between the operators entering into the operator product expansions of the structure functions. The constructed operator bases, which include the powers of the dissipation operator and the enstrophy operator, provide the possibility to calculate the exponents of the anomalous scaling.Comment: 9 pages, LaTeX2e(iopart.sty), submitted to J. Phys. A: Math. Ge