We show that stable bright solitons can appear in a medium with spatially inhomogeneous self-defocusing (SDF) nonlinearity of a double-well structure. For a specific choice of the nonlinearity parameters, we obtain exact analytical solutions for the fundamental bright solitons. By making use of the linear stability analysis, the stability region in the parameter space for the exact fundamental bright soliton is obtained numerically. We also show the bifurcation from an antisymmetric to an asymmetric bright soliton for the SDF double-well nonlinearity.

In this paper, which constitutes the first part of the series, we consider calculation of two-center Coulomb and hybrid integrals over Slater-type orbitals. General formulas for these integrals are derived with no restrictions on the values of the quantum numbers and nonlinear parameters. Direct integration over the coordinates of one of the electrons leaves us with the set of overlaplike integrals which are evaluated by using two distinct methods. The first one is based on the transformation to the ellipsoidal coordinates system and the second utilizes a recursive scheme for consecutive increase of the angular momenta in the integrand. In both methods simple one-dimensional numerical integrations are used in order to avoid severe digital erosion connected with the straightforward use of the alternative analytical formulas. It is discussed that the numerical integration does not introduce a large computational overhead since the integrands are well-behaved functions, calculated recursively with decent speed. Special attention is paid to the numerical stability of the algorithms. Applicability of the resulting scheme over a large range of the nonlinear parameters is tested on examples of the most difficult integrals appearing in the actual calculations including, at most, $7i$-type functions $\left(l=6\right)$.

In this paper we consider the calculation of two-center exchange integrals over Slater-type orbitals (STOs). We apply the Neumann expansion of the Coulomb interaction potential and consider calculation of all basic quantities which appear in the resulting expression. Analytical closed-form equations for all auxiliary quantities have already been known but they suffer from large digital erosion when some of the parameters are large or small. We derive two differential equations which are obeyed by the most difficult basic integrals. Taking them as a starting point, useful series expansions for small parameter values or asymptotic expansions for large parameter values are systematically derived. The resulting expansions replace the corresponding analytical expressions when the latter introduce significant cancellations. Additionally, we reconsider numerical integration of some necessary quantities and present a new way to calculate the integrand with a controlled precision. All proposed methods are combined to lead to a general, stable algorithm. We perform extensive numerical tests of the introduced expressions to verify their validity and usefulness. Advances reported here provide methodology to compute two-electron exchange integrals over STOs for a broad range of the nonlinear parameters and large angular momenta.

In the rotational sandpile model, either the clockwise or the anticlockwise toppling rule is assigned to all the lattice sites. It has all the features of a stochastic sandpile model but belongs to a different universality class than the Manna class. A crossover from rotational to Manna universality class is studied by constructing a random rotational sandpile model and assigning randomly clockwise and anticlockwise rotational toppling rules to the lattice sites. The steady state and the respective critical behavior of the present model are found to have a strong and continuous dependence on the fraction of the lattice sites having the anticlockwise (or clockwise) rotational toppling rule. As the anticlockwise and clockwise toppling rules exist in equal proportions, it is found that the model reproduces critical behavior of the Manna model. It is then further evidence of the existence of the Manna class, in contradiction with some recent observations of the nonexistence of the Manna class.

In this paper, we provide a scheme to compute the absolute free energy of a smectic-$A$ phase via the “indirect method.” The state of interest is connected through a three-step reversible path to a reference state. This state consists of a low-density layer of rods coupled to two external fields maintaining these rods close to the layer's plane and oriented preferably normal to the layer. The low-density free energy of the reference state can be computed on the basis of the relevant second virial coefficients between two rods coupled to the two external fields. We apply this technique to the Gay-Berne potential for calamitics with a parameter set leading to stable isotropic (I), nematic (N), smectic-$A$ (SmA), and crystal (Cr) phases. We locate the I-SmA phase transition at low pressure and the sequence of phase transitions I-N-SmA along higher-pressure isobars and we establish the location of the I-N-SmA triple point. Close to this triple point, we show that the N-SmA transition is clearly first order. Our results are compared to the coexistence lines of the approximate phase diagram elucidated by de Miguel et al. [J. Chem. Phys. 121, 11183 (2004)] established through the direct observation of the sequence of phase transitions occurring along isobars under heating or cooling sequences of runs. Finally, we discuss the potential of our technique in studying similar transitions observed on layered phases under confinement.

The origin of linear instability resulting in rotating sheared accretion flows has remained a controversial subject for a long time. While some explanations of such non-normal transient growth of disturbances in the Rayleigh stable limit were available for magnetized accretion flows, similar instabilities in the absence of magnetic perturbations remained unexplained. This dichotomy was resolved in two recent publications by Chattopadhyay and co-workers [Mukhopadhyay and Chattopadhyay, J. Phys. A 46, 035501 (2013); Nath et al., Phys. Rev. E 88, 013010 (2013)] where it was shown that such instabilities, especially for nonmagnetized accretion flows, were introduced through interaction of the inherent stochastic noise in the system (even a “cold” accretion flow at 3000 K is too “hot” in the statistical parlance and is capable of inducing strong thermal modes) with the underlying Taylor-Couette flow profiles. Both studies, however, excluded the additional energy influx (or efflux) that could result from nonzero cross correlation of a noise perturbing the velocity flow, say, with the noise that is driving the vorticity flow (or equivalently the magnetic field and magnetic vorticity flow dynamics). Through the introduction of such a time symmetry violating effect, in this article we show that nonzero noise cross correlations essentially renormalize the strength of temporal correlations. Apart from an overall boost in the energy rate (both for spatial and temporal correlations, and hence in the ensemble averaged energy spectra), this results in mutual competition in growth rates of affected variables often resulting in suppression of oscillating Alfven waves at small times while leading to faster saturations at relatively longer time scales. The effects are seen to be more pronounced with magnetic field fluxes where the noise cross correlation magnifies the strength of the field concerned. Another remarkable feature noted specifically for the autocorrelation functions is the removal of energy degeneracy in the temporal profiles of fast growing non-normal modes leading to faster saturation with minimum oscillations. These results, including those presented in the previous two publications, now convincingly explain subcritical transition to turbulence in the linear limit for all possible situations that could now serve as the benchmark for nonlinear stability studies in Keplerian accretion disks.

The inverse of the von Kármán constant $\kappa$ is the leading coefficient in the equation describing the logarithmic mean velocity profile in wall bounded turbulent flows. Klewicki [J. Fluid Mech. 718, 596 (2013)] connects the asymptotic value of $\kappa$ with an emerging condition of dynamic self-similarity on an interior inertial domain that contains a geometrically self-similar hierarchy of scaling layers. A number of properties associated with the asymptotic value of $\kappa$ are revealed. This is accomplished using a framework that retains connection to invariance properties admitted by the mean statement of dynamics. The development leads toward, but terminates short of, analytically determining a value for $\kappa$. It is shown that if adjacent layers on the hierarchy (or their adjacent positions) adhere to the same self-similarity that is analytically shown to exist between any given layer and its position, then $\kappa \equiv {\Phi }^{-2}=0.381966...$, where $\Phi =(1+\sqrt{5})/2$ is the golden ratio. A number of measures, derived specifically from an analysis of the mean momentum equation, are subsequently used to empirically explore the veracity and implications of $\kappa ={\Phi }^{-2}$. Consistent with the differential transformations underlying an invariant form admitted by the governing mean equation, it is demonstrated that the value of $\kappa$ arises from two geometric features associated with the inertial turbulent motions responsible for momentum transport. One nominally pertains to the shape of the relevant motions as quantified by their area coverage in any given wall-parallel plane, and the other pertains to the changing size of these motions in the wall-normal direction. In accord with self-similar mean dynamics, these two features remain invariant across the inertial domain. Data from direct numerical simulations and higher Reynolds number experiments are presented and discussed relative to the self-similar geometric structure indicated by the analysis, and in particular the special form of self-similarity shown to correspond to $\kappa ={\Phi }^{-2}$.

We study the orientational ordering in systems of self-propelled particles with selective interactions. To introduce the selectivity we augment the standard Vicsek model with a bounded-confidence collision rule: a given particle only aligns to neighbors who have directions quite similar to its own. Neighbors whose directions deviate more than a fixed restriction angle $\alpha$ are ignored. The collective dynamics of this system is studied by agent-based simulations and kinetic mean-field theory. We demonstrate that the reduction of the restriction angle leads to a critical noise amplitude decreasing monotonically with that angle, turning into a power law with exponent $\frac{3}{2}$ for small angles. Moreover, for small system sizes we show that upon decreasing the restriction angle, the kind of the transition to polar collective motion changes from continuous to discontinuous. Thus, an apparent tricritical point with different scaling laws is identified and calculated analytically. We investigate the shifting and vanishing of this point due to the formation of density bands as the system size is increased. Agent-based simulations in small systems with large particle velocities show excellent agreement with the kinetic theory predictions. We also find that at very small interaction angles, the polar ordered phase becomes unstable with respect to the apolar phase. We derive analytical expressions for the dependence of the threshold noise on the restriction angle. We show that the mean-field kinetic theory also permits stationary nematic states below a restriction angle of $0.681\pi$. We calculate the critical noise, at which the disordered state bifurcates to a nematic state, and find that it is always smaller than the threshold noise for the transition from disorder to polar order. The disordered-nematic transition features two tricritical points: At low and high restriction angle, the transition is discontinuous but continuous at intermediate $\alpha$. We generalize our results to systems that show fragmentation into more than two groups and obtain scaling laws for the transition lines and the corresponding tricritical points. A numerical method to evaluate the nonlinear Fredholm integral equation for the stationary distribution function is also presented. This method is shown to give excellent agreement with agent-based simulations, even in strongly ordered systems at noise values close to zero.

We show that the recently proposed correlated sampling without reweighting procedure extends the locality (asymptotic independence of the system size) of a physical property to the statistical fluctuations of its estimator. This makes the approach potentially vastly more efficient for computing space-localized properties in large systems compared with standard correlated methods. A proof is given for a large collection of noninteracting fragments. Calculations on hydrogen chains suggest that this behavior holds not only for systems displaying short-range correlations, but also for systems with long-range correlations.

Motion of particles in many systems exhibits a mixture between periods of random diffusive-like events and ballistic-like motion. In many cases, such systems exhibit strong anomalous diffusion, where low-order moments $\langle |x\left(t\right){|}^q\rangle$ with $q$ below a critical value $q_c$ exhibit diffusive scaling while for $q>q_c$ a ballistic scaling emerges. The mixed dynamics constitutes a theoretical challenge since it does not fall into a unique category of motion, e.g., the known diffusion equations and central limit theorems fail to describe both aspects. In this paper we resolve this problem by resorting to the concept of infinite density. Using the widely applicable Lévy walk model, we find a general expression for the corresponding non-normalized density which is fully determined by the particles velocity distribution, the anomalous diffusion exponent $\alpha$, and the diffusion coefficient $K_{\alpha }$. We explain how infinite densities play a central role in the description of dynamics of a large class of physical processes and discuss how they can be evaluated from experimental or numerical data.

Is the brain really operating at a critical point? We study the nonequilibrium properties of a neural network which models the dynamics of the neocortex and argue for optimal quasicritical dynamics on the Widom line where the correlation length and information transmission are optimized. We simulate the network and introduce an analytical mean-field approximation, characterize the nonequilibrium phase transitions, and present a nonequilibrium phase diagram, which shows that in addition to an ordered and disordered phase, the system exhibits a “quasiperiodic” phase corresponding to synchronous activity in simulations, which may be related to the pathological synchronization associated with epilepsy.

The biological function of a protein often depends on the formation of an ordered structure in order to support a smaller, chemically active configuration of amino acids against thermal fluctuations. Here we explore the development of proteins evolving to satisfy this requirement using an off-lattice polymer model in which monomers interact as low resolution amino acids. To evolve the model, we construct a Markov process in which sequences are subjected to random replacements, insertions, and deletions and are selected to recover a predefined minimum number of solid-ordered monomers using the Lindemann melting criterion. We show that polymers generated by this process consistently fold into soluble, ordered globules of similar length and complexity to small protein motifs. To compare the evolution of the globules with proteins, we analyze the statistics of amino acid replacements, the dependence of site mutation rates on solvent exposure, and the dependence of structural distance on sequence distance for homologous alignments. Despite the simplicity of the model, the results display a surprisingly close correspondence with protein data.

When modeling of tumor-driven angiogenesis, a major source of analytical and computational complexity is the strong coupling between the kinetic parameters of the relevant stochastic branching-and-growth of the capillary network, and the family of interacting underlying fields. To reduce this complexity, we take advantage of the system intrinsic multiscale structure: we describe the stochastic dynamics of the cells at the vessel tip at their natural mesoscale, whereas we describe the deterministic dynamics of the underlying fields at a larger macroscale. Here, we set up a conceptual stochastic model including branching, elongation, and anastomosis of vessels and derive a mean field approximation for their densities. This leads to a deterministic integropartial differential system that describes the formation of the stochastic vessel network. We discuss the proper capillary injecting boundary conditions and include the results of relevant numerical simulations.

Dynamical systems based on the minority game (MG) have been a paradigm for gaining significant insights into a variety of social and biological behaviors. Recently, a grouping phenomenon has been unveiled in MG systems of multiple resources (strategies) in which the strategies spontaneously break into an even number of groups, each exhibiting an identical oscillation pattern in the attendance of game players. Here we report our finding of spontaneous breakup of resources into three groups, each exhibiting period-three oscillations. An analysis is developed to understand the emergence of the striking phenomenon of triple grouping and period-three oscillations. In the presence of random disturbances, the triple-group/period-three state becomes transient, and we obtain explicit formula for the average transient lifetime using two methods of approximation. Our finding indicates that, period-three oscillation, regarded as one of the most fundamental behaviors in smooth nonlinear dynamical systems, can also occur in much more complex, evolutionary-game dynamical systems. Our result also provides a plausible insight for the occurrence of triple grouping observed, for example, in the U.S. housing market.

In this work, we investigate the dynamics of an uniaxial silica fiber under the viewpoint of propagation of ultimately ultrashort optical waveguide channels. As a result, we unveil the existence of three typical kinds of ultrabroadband excitations whose profiles strongly depend upon their angular momenta. Looking forward to surveying their scattering features, we unearth some underlying head-on scenarios of elastic collisions. Accordingly, we address some useful and straightforward applications in nonlinear optics through secured data transmission systems, as well as laser physics and soliton theory with optical soliton dynamics.

We consider three-dimensional reshaping of a thin nematoelastic film upon nematic-isotropic transition in the field of a charge one topological defect, leading to either cone or anticone ($d$-cone) shells. The analysis is based on the relation between the shell metric and the tensor order parameter under the assumption of no elastic deformation and volume change. The shape of the shell can be modified by doping, creating cones with curved generatrices. Anticones necessarily have an even number of radial creases. The curvature singularity at the apex is resolved due to decay of the nematic order parameter at the defect core.

Liquid crystals in two dimensions do not support long-range nematic order, but a quasinematic phase where the orientational correlations decay algebraically is possible. The transition from the isotropic to the quasinematic phase can be continuous and of the Kosterlitz-Thouless type, or it can be first order. We report here on a liquid-crystal model where the nature of the isotropic to quasinematic transition can be tuned via a single parameter $p$ in the pair potential. For $p<p_{\mathrm{t}}$, the transition is of the Kosterlitz-Thouless type, while for $p>p_{\mathrm{t}}$, it is first order. Precisely at $p=p_{\mathrm{t}}$, there is a tricritical point where, in addition to the orientational correlations, also the positional correlations decay algebraically. The tricritical behavior is analyzed in detail, including an accurate estimate of $p_{\mathrm{t}}$. The results follow from extensive Monte Carlo simulations combined with a finite-size scaling analysis. Paramount in the analysis is a scheme to facilitate the extrapolation of simulation data in parameters that are not necessarily field variables (in this case, the parameter $p$), the details of which are also provided. This scheme provides a simple and powerful alternative for situations where standard histogram reweighting cannot be applied.

We study the anisotropic Kardar-Parisi-Zhang equation using nonperturbative renormalization group methods. In contrast to a previous analysis in the weak-coupling regime, we find the strong-coupling fixed point corresponding to the isotropic rough phase to be always locally stable and unaffected by the anisotropy even at noninteger dimensions. Apart from the well-known weak-coupling and the now well-established isotropic strong-coupling behavior, we find an anisotropic strong-coupling fixed point for nonlinear couplings of opposite signs at noninteger dimensions.

We consider the efficiency at maximum power of a quantum Otto engine, which uses a spin or a harmonic system as its working substance and works between two heat reservoirs at constant temperatures $T_h$ and $T_c$ $(<T_h)$. Although the behavior of spin-$1/2$ system differs substantially from that of the harmonic system in that they obey two typical quantum statistics, the efficiencies at maximum power based on these two different kinds of quantum systems are bounded from the upper side by the same expression ${\eta }_{\mathrm{mp}}\le {\eta }_{+}\equiv {\eta }_C^2/[{\eta }_C-(1-{\eta }_C)ln(1-{\eta }_C)]$ with ${\eta }_C=1-T_c/T_h$ as the Carnot efficiency. This expression ${\eta }_{\mathrm{mp}}$ possesses the same universality of the CA efficiency ${\eta }_{\mathrm{CA}}=1-\sqrt{1-{\eta }_C}$ at small relative temperature difference. Within the context of irreversible thermodynamics, we calculate the Onsager coefficients and show that the value of ${\eta }_{\mathrm{CA}}$ is indeed the upper bound of EMP for an Otto engine working in the linear-response regime.