Renormalization Group Approach to the Infrared Behavior of a Zero-Temperature Bose System

We exploit the renormalization-group approach to establish the {\em exact} infrared behavior of an interacting Bose system at zero temperature. The local-gauge symmetry in the broken-symmetry phase is implemented through the associated Ward identities, which reduce the number of independent running couplings to a single one. For this coupling the $\epsilon$-expansion can be controlled to all orders in $\epsilon$ ($=3-d$). For spatial dimensions $1 < d \leq 3$ the Bogoliubov fixed point is unstable towards a different fixed point characterized by the divergence of the longitudinal correlation function. The Bogoliubov linear spectrum, however, is found to be independent from the critical behavior of this correlation function, being exactly constrained by Ward identities. The new fixed point properly gives a finite value of the coupling among transverse fluctuations, but due to virtual intermediate longitudinal fluctuations the effective coupling affecting the transverse correlation function flows to zero. As a result, no transverse anomalous dimension is present. This treatment allows us to recover known results for the quantum Bose gas in the context of a unifying framework and also to reveal the non-trivial skeleton structure of its perturbation theory.Comment: 21 page

Vortex -- Kink Interaction and Capillary Waves in a Vector Superfluid

Interaction of a vortex in a circularly polarized superfluid component of a 2d complex vector field with the phase boundary between superfluid phases with opposite signs of polarization leads to a resonant excitation of a ``capillary'' wave on the boundary. This leads to energy losses by the vortex--image pair that has to cause its eventual annihilation.Comment: LaTeX 7 pages, no figure

Weakly non-linear dynamics in reaction -- diffusion systems with L\'{e}vy flights

Reaction--diffusion equations with a fractional Laplacian are reduced near a long wave Hopf bifurcation. The obtained amplitude equation is shown to be the complex Ginzburg-Landau equation with a fractional Laplacian. Some of the properties of the normal complex Ginzburg-Landau equation are generalised for the fractional analogue. In particular, an analogue of Kuramoto-Sivashinsky equation is derived

Non-homogeneous random walks, subdiffusive migration of cells and anomalous chemotaxis

This paper is concerned with a non-homogeneous in space and non-local in time random walk model for anomalous subdiffusive transport of cells. Starting with a Markov model involving a structured probability density function, we derive the non-local in time master equation and fractional equation for the probability of cell position. We show the structural instability of fractional subdiffusive equation with respect to the partial variations of anomalous exponent. We find the criteria under which the anomalous aggregation of cells takes place in the semi-infinite domain.Comment: 18 pages, accepted for publicatio

Domain walls and vortices in linearly coupled systems

We investigate 1D and 2D radial domain-wall (DW) states in the system of two nonlinear-Schr\"{o}dinger/Gross-Pitaevskii equations, which are coupled by the linear mixing and by the nonlinear XPM (cross-phase-modulation). The system has straightforward applications to two-component Bose-Einstein condensates, and to the bimodal light propagation in nonlinear optics. In the former case, the two components represent different hyperfine atomic states, while in the latter setting they correspond to orthogonal polarizations of light. Conditions guaranteeing the stability of flat continuous wave (CW) asymmetric bimodal states are established, followed by the study of families of the corresponding DW patterns. Approximate analytical solutions for the DWs are found near the point of the symmetry-breaking bifurcation of the CW states. An exact DW solution is produced for ratio 3:1 of the XPM and SPM coefficients. The DWs between flat asymmetric states, which are mirror images to each other, are completely stable, and all other species of the DWs, with zero crossings in one or two components, are fully unstable. Interactions between two DWs are considered too, and an effective potential accounting for the attraction between them is derived analytically. Direct simulations demonstrate merger and annihilation of the interacting DWs. The analysis is extended for the system including single- and double-peak external potentials. Generic solutions for trapped DWs are obtained in a numerical form, and their stability is investigated. An exact stable solution is found for the DW trapped by a single-peak potential. In the 2D geometry, stable two-component vortices are found, with topological charges s=1,2,3. Radial oscillations of annular DW-shaped pulsons, with s=0,1,2, are studied too. A linear relation between the period of the oscillations and the mean radius of the DW ring is derived analytically.Comment: Physical Review E, in pres

Oscillatory instability in super-diffusive reaction -- diffusion systems: fractional amplitude and phase diffusion equations

Nonlinear evolution of a reaction--super-diffusion system near a Hopf bifurcation is studied. Fractional analogues of complex Ginzburg-Landau equation and Kuramoto-Sivashinsky equation are derived, and some of their analytical and numerical solutions are studied

Stable two-dimensional solitary pulses in linearly coupled dissipative Kadomtsev-Petviashvili equations

A two-dimensional (2D) generalization of the stabilized Kuramoto - Sivashinsky (KS) system is presented. It is based on the Kadomtsev-Petviashvili (KP) equation including dissipation of the generic (Newell -- Whitehead -- Segel, NWS) type and gain. The system directly applies to the description of gravity-capillary waves on the surface of a liquid layer flowing down an inclined plane, with a surfactant diffusing along the layer's surface. Actually, the model is quite general, offering a simple way to stabilize nonlinear waves in media combining the weakly-2D dispersion of the KP type with gain and NWS dissipation. Parallel to this, another model is introduced, whose dissipative terms are isotropic, rather than of the NWS type. Both models include an additional linear equation of the advection-diffusion type, linearly coupled to the main KP-NWS equation. The extra equation provides for stability of the zero background in the system, opening a way to the existence of stable localized pulses. The consideration is focused on the case when the dispersive part of the system of the KP-I type, admitting the existence of 2D localized pulses. Treating the dissipation and gain as small perturbations and making use of the balance equation for the field momentum, we find that the equilibrium between the gain and losses may select two 2D solitons, from their continuous family existing in the conservative counterpart of the model (the latter family is found in an exact analytical form). The selected soliton with the larger amplitude is expected to be stable. Direct simulations completely corroborate the analytical predictions.Comment: a latex text file and 16 eps files with figures; Physical Review E, in pres

Nonadiabatic Dynamics of Atoms in Nonuniform Magnetic Fields

Dynamics of neutral atoms in nonuniform magnetic fields, typical of quadrupole magnetic traps, is considered by applying an accurate method for solving nonlinear systems of differential equations. This method is more general than the adiabatic approximation and, thus, permits to check the limits of the latter and also to analyze nonadiabatic regimes of motion. An unusual nonadiabatic regime is found when atoms are confined from one side of the z-axis but are not confined from another side. The lifetime of atoms in a trap in this semi-confining regime can be sufficiently long for accomplishing experiments with a cloud of such atoms. At low temperature, the cloud is ellipsoidal being stretched in the axial direction and moving along the z-axis. The possibility of employing the semi-confining regime for studying the relative motion of one component through another, in a binary mixture of gases is discussed.Comment: 1 file, 17 pages, RevTex, 2 table

Linear Stability of Fractional Reaction - Diffusion Systems

Theoretical framework for linear stability of an anomalous sub-diffusive activator-inhibitor system is set. Generalized Turing instability conditions are found to depend on anomaly exponents of various species. In addition to monotonous instability, known from normal diffusion, in an anomalous system oscillatory modes emerge. For equal anomaly exponents for both species the type of unstable modes is determined by the ratio of the reactants' diffusion coefficients. When the ratio exceeds its normal critical value, the monotonous modes become stable, whereas oscillatory instability persists until the anomalous critical value is also exceeded. An exact formula for the anomalous critical value is obtained. It is shown that in the short wave limit the growth rate is a power law of the wave number. When the anomaly exponents differ, disturbance modes are governed by power laws of the distinct exponents. If the difference between the diffusion anomaly exponents is small, the splitting of the power law exponents is absent at the leading order and emerges only as a next-order effect. In the short wave limit the onset of instability is governed by the anomaly exponents, whereas the ratio of diffusion coefficients influences the complex growth rates. When the exponent of the inhibitor is greater than that of the activator, the system is always unstable due to the inhibitor enhanced diffusion relatively to the activator. If the exponent of the activator is greater, the system is always stable. Existence of oscillatory unstable modes is also possible for waves of moderate length

Low-Dimensional Description of Pulses under the Action of Global Feedback Control

The influence of a global delayed feedback control which acts on a system governed by a subcritical complex Ginzburg-Landau equation is considered. The method based on a variational principle is applied for the derivation of low-dimensional evolution models. In the framework of those models, one-pulse and two-pulse solutions are found, and their linear stability analysis is carried out. The application of the finite-dimensional model allows to reveal the existence of chaotic oscillatory regimes and regimes with double-period and quadruple-period oscillations. The diagram of regimes resembles those found in the damped-driven nonlinear Schrödinger equation. The obtained results are compared with the results of direct numerical simulations of the original problem