We study localized nonlinear excitations in diffusive Hindmarsh-Rose neural networks. We show that the Hindmarsh-Rose model can be reduced to a modified Complex Ginzburg-Landau equation through the application of a perturbation technique. We equally report on the presence of envelop solitons of the nerve impulse in this neural network. From the biological point of view, this result suggests that neurons can participate in a collective processing of information, a relevant part of which is shared over all neurons but not concentrated at the single neuron level. By employing the standard linear stability analysis, the growth rate of the modulational instability is derived as a function of the wave number and systems parameters.

An efficient way of evolving a solution to an ordinary differential equation is presented. A finite element method is used where we expand in a convenient local basis set of functions that enforce both function and first derivative continuity across the boundaries of each element. We also implement an adaptive step-size choice for each element that is based on a Taylor series expansion. This algorithm is used to solve for the eigenpairs corresponding to the one-dimensional soft Coulomb potential, $1/\sqrt{x^2+{\beta }^2}$, which becomes numerically intractable (because of extreme stiffness) as the softening parameter ($\beta$) approaches zero. We are able to maintain near machine accuracy for $\beta$ as low as $\beta ={10}^{-8}$ using 16-digit precision calculations. Our numerical results provide insight into the controversial one-dimensional hydrogen atom, which is a limiting case of the soft Coulomb problem as $\beta \to 0$.

In this paper, a phase-field-based multiple-relaxation-time lattice Boltzmann (LB) model is proposed for incompressible multiphase flow systems. In this model, one distribution function is used to solve the Chan-Hilliard equation and the other is adopted to solve the Navier-Stokes equations. Unlike previous phase-field-based LB models, a proper source term is incorporated in the interfacial evolution equation such that the Chan-Hilliard equation can be derived exactly and also a pressure distribution is designed to recover the correct hydrodynamic equations. Furthermore, the pressure and velocity fields can be calculated explicitly. A series of numerical tests, including Zalesak's disk rotation, a single vortex, a deformation field, and a static droplet, have been performed to test the accuracy and stability of the present model. The results show that, compared with the previous models, the present model is more stable and achieves an overall improvement in the accuracy of the capturing interface. In addition, compared to the single-relaxation-time LB model, the present model can effectively reduce the spurious velocity and fluctuation of the kinetic energy. Finally, as an application, the Rayleigh-Taylor instability at high Reynolds numbers is investigated.

Infinite arrays of coupled two-state stochastic oscillators exhibit well-defined steady states. We study the fluctuations that occur when the number $N$ of oscillators in the array is finite. We choose a particular form of global coupling that in the infinite array leads to a pitchfork bifurcation from a monostable to a bistable steady state, the latter with two equally probable stationary states. The control parameter for this bifurcation is the coupling strength. In finite arrays these states become metastable: The fluctuations lead to distributions around the most probable states, with one maximum in the monostable regime and two maxima in the bistable regime. In the latter regime, the fluctuations lead to transitions between the two peak regions of the distribution. Also, we find that the fluctuations break the symmetry in the bimodal regime, that is, one metastable state becomes more probable than the other, increasingly so with increasing array size. To arrive at these results, we start from microscopic dynamical evolution equations from which we derive a Langevin equation that exhibits an interesting multiplicative noise structure. We also present a master equation description of the dynamics. Both of these equations lead to the same Fokker-Planck equation, the master equation via a $1/N$ expansion and the Langevin equation via standard methods of Itô calculus for multiplicative noise. From the Fokker-Planck equation we obtain an effective potential that reflects the transition from the monomodal to the bimodal distribution as a function of a control parameter. We present a variety of numerical and analytic results that illustrate the strong effects of the fluctuations. We also show that the limits $N\to \infty$ and $t\to \infty$ ($t$ is the time) do not commute. In fact, the two orders of implementation lead to drastically different results.

Using Monte Carlo simulation, we analyze phase transitions of two antiferromagnetic (AFM) triangular Blume-Capel (BC) models with AFM interactions between third-nearest neighbors. One model has hard-core exclusions between the nearest-neighbor (1NN) particles (3NN1 model) and the other has them between the nearest-neighbor and next-nearest-neighbor particles (3NN12 model). Finite-size scaling analysis reveals that in these models, the transition from the paramagnetic to long-range order (LRO) AFM phase is either of the first order or goes through an intermediate phase which might be attributed to the Berezinskii-Kosterlitz-Thouless (BKT) type. The properties of the low-temperature phase transition to the AFM phase of the 1NN, 3NN1, and 3NN12 models are found to be very similar for almost all values of a normalized single-ion anisotropy parameter, $0<\delta <1.5$. Higher temperature behavior of the 3NN12 and 3NN1 models is rather different from that of the 1NN model. Three phase transitions are observed for the 3NN12 model: from the paramagnetic phase to the phase with domains of the LRO AFM phase at $T_c$, from this structure to the diluted frustrated BKT-type phase at $T_2$, and from the frustrated phase to the AFM LRO phase at $T_1$. For the 3NN12 model, $T_c>T_2>T_1$ at $0<\delta <1.15$ (range I), $T_c\approx T_2>T_1$ at $1.15<\delta <1.3$ (range II), and $T_c=T_2=T_1$ at $1.3<\delta <1.5$ (range III). For the 3NN1 model, $T_c\approx T_2>T_1$ at $0<\delta <1.2$ (range II) and $T_c=T_2=T_1$ at $1.2<\delta <1.5$ (range III). There is only one first-order phase transition in range III. The transition at $T_c$ is of the first order in range II and either of a weak first order or a second order in range I.

In the literature, it is pointed out that non-Brownian particles tend to show shear-induced diffusive behavior due to hydrodynamic interactions. Several authors indicate a long correlation time of the particle velocities in comparison to Brownian particle velocities modeled by a white noise. This work deals with the derivation of a Fokker-Planck equation both in position and velocity space which describes the process of shear-induced self-diffusion, whereas, so far, this problem has been described by Fokker-Planck equations restricted to position space. The long velocity correlation times actually would necessitate large time-step sizes in the mathematical description of the problem in order to capture the diffusive regime. In fact, time steps of specific lengths pose problems to the derivation of the corresponding Fokker-Planck equation because the whole particle configuration changes during long time-step sizes. On the other hand, small time-step sizes, i.e., in the range of the velocity correlation time, violate the Markov property of the position variable. In this work we regard the problem of shear-induced self-diffusion with respect to the Markov property and reformulate the problem with respect to small time-step sizes. In this derivation, we regard the nondimensionalized Langevin equation and develop a new compact form which allows us to analyze the Langevin equation for all time scales of interest for both Brownian and non-Brownian particles starting from a single equation. This shows that the Fokker-Planck equation in position space should be extended to a colored-noise Fokker-Planck equation in both position and colored-noise velocity space, which we will derive.

Network representations are useful for describing the structure of a large variety of complex systems. Although most studies of real-world networks suppose that nodes are connected by only a single type of edge, most natural and engineered systems include multiple subsystems and layers of connectivity. This new paradigm has attracted a great deal of attention and one fundamental challenge is to characterize multilayer networks both structurally and dynamically. One way to address this question is to study the spectral properties of such networks. Here we apply the framework of graph quotients, which occurs naturally in this context, and the associated eigenvalue interlacing results to the adjacency and Laplacian matrices of undirected multilayer networks. Specifically, we describe relationships between the eigenvalue spectra of multilayer networks and their two most natural quotients, the network of layers and the aggregate network, and show the dynamical implications of working with either of the two simplified representations. Our work thus contributes in particular to the study of dynamical processes whose critical properties are determined by the spectral properties of the underlying network.

I propose an index for evaluating the credibility of confidence intervals for future observables predicted from high-dimensional time-series data. The index evaluates the distance from the current state to the data manifold. I demonstrate the index with artificial datasets generated from the Lorenz'96 II model [Lorenz, in Proceedings of the Seminar on Predictability, Vol. 1 (ECMWF, Reading, UK, 1996), p. 1], the Lorenz'96 I model [Hansen and Smith, J. Atmos. Sci. 57, 2859 (2000)], and the coupled map lattice, and a real dataset for the solar irradiation around Japan.

We investigate the dynamical instability of Bose-Einstein condensates (BECs) with higher-order interactions immersed in an optical lattice with weak driving harmonic potential. For this, we compute both analytically and numerically a modified Gross-Pitaevskii equation with higher-order nonlinearity and external potentials generated by magnetic and optical fields. Using the time-dependent variational approach, we derive the ordinary differential equations for the time evolution of the amplitude and phase of modulational perturbation. Through an effective potential, we obtain the modulational instability condition of BECs and discuss the effect of the higher-order interaction in the dynamics of the condensates in presence of optical potential. We perform direct numerical simulations to support our analytical results, and good agreement is found.

We consider the filling of a capillary by a viscoelastic fluid described by the Phan-Thien–Tanner (PTT) constitutive behavior. By considering both vertical capillary filling and horizontal capillary filling, we demarcate the role played by gravity and fluid rheology towards long-time oscillations in the capillary penetration depth. We also consider the isothermal filling of the capillary for a closed channel and thus bring out the fundamental differences in the nature of capillary filling for PTT and Newtonian fluids for closed channels in comparison to open channels. Through a scaling analysis, we highlight a distinct viscoelastic regime in the horizontal capillary filling which is in contrast to the Washburn scaling seen in the case of Newtonian fluids. Such an analysis with a very general constitutive behavior is therefore expected to shed light on many areas of microfluidics which focus on biofluids that are often well described by the PTT constitutive behavior.

The work of Spitzer on dynamical friction in a plasma [L. Spitzer, Jr., Physics of Fully Ionized Gases, 2nd ed. (Wiley, New York, 1962), Chap. 5] is extended to relativistic systems. We derive the force of dynamical friction, diffusion tensor, and test particle relaxation rates for a Maxwellian background in the same form as Trubnikov [B. A. Trubnikov, in Reviews of Plasma Physics, edited by M. A. Leontovich (Consultants Bureau, New York, 1965), Vol. 1, p. 105], enabling high-temperature laboratory and astrophysical plasmas to be modeled in a consistent manner.

We investigate the accuracy and performance of the regularized version of the single-relaxation-time lattice Boltzmann equation for the case of two- and three-dimensional lid-driven cavities. The regularized version is shown to provide a significant gain in stability over the standard single-relaxation time, at a moderate computational overhead.

We investigate the self-intermediate scattering function (SISF) in a three-dimensional (3D) cubic lattice fluid (interacting lattice gas) with attractive nearest-neighbor interparticle interactions at a temperature slightly above the critical one by means of Monte Carlo simulations. A special representation of SISF as an exponent of the mean tracer diffusion coefficient multiplied by the geometrical factor and time is considered to highlight memory effects that are included in time and wave-vector dependence of the diffusion coefficient. An analytical expression for the diffusion coefficient is suggested to reproduce the simulation data. It is shown that the particles' mean-square displacement is equal to the time integral of the diffusion coefficient. We make a comparison with the previously considered 2D system on a square lattice. The main difference with the two-dimensional case is that the time dependence of particular characteristics of the tracer diffusion coefficient in the 3D case cannot be described by exponentially decreasing functions, but requires using stretched exponentials with rather small values of exponents, of the order of 0.2. The hydrodynamic values of the tracer diffusion coefficient (in the limit of large times and small wave vectors) defined through SIFS simulation results agree well with the results of its direct determination by the mean-square displacement of the particles in the entire range of concentrations and temperatures.

Solutions for the continuous-time random walk (CTRW) model are known in few cases. In this work, the uncoupled CTRW model is investigated analytically and numerically. In particular, the probability density function (PDF) and $n$-moment are obtained and analyzed. Exponential and Gaussian functions are used for the jump length PDF, whereas the Mittag-Leffler function and a combination of exponential and power-laws function is used for the waiting time PDF. The exponential and Gaussian jump length PDFs have finite jump length variances and they give the same second moment; however, their distribution functions present different behaviors near the origin. The combination of exponential and power-law function for the waiting time PDF can generate a crossover from anomalous regime to normal regime. Moreover, the parameter of the exponential jump length PDF does not change the behavior of the $n$-moment for all time intervals, and for the Gaussian jump length PDF the $n$-moment also indicates a similar behavior.

The Weibull distribution is a commonly used model for the strength of brittle materials and earthquake return intervals. Deviations from Weibull scaling, however, have been observed in earthquake return intervals and the fracture strength of quasibrittle materials. We investigate weakest-link scaling in finite-size systems and deviations of empirical return interval distributions from the Weibull distribution function. Our analysis employs the ansatz that the survival probability function of a system with complex interactions among its units can be expressed as the product of the survival probability functions for an ensemble of representative volume elements (RVEs). We show that if the system comprises a finite number of RVEs, it obeys the $\kappa$-Weibull distribution. The upper tail of the $\kappa$-Weibull distribution declines as a power law in contrast with Weibull scaling. The hazard rate function of the $\kappa$-Weibull distribution decreases linearly after a waiting time ${\tau }_c\propto n^{1/m}$, where $m$ is the Weibull modulus and $n$ is the system size in terms of representative volume elements. We conduct statistical analysis of experimental data and simulations which show that the $\kappa$ Weibull provides competitive fits to the return interval distributions of seismic data and of avalanches in a fiber bundle model. In conclusion, using theoretical and statistical analysis of real and simulated data, we demonstrate that the $\kappa$-Weibull distribution is a useful model for extreme-event return intervals in finite-size systems.

One of the central themes in modern ecology is the enduring debate on whether there is a relationship between the complexity of a biological community and its stability. In this paper, we focus on the role of detritus and spatial dispersion on the stability of ecosystems. Using Monte Carlo simulations we analyze two three-level models of food webs: a grazing one with the basal species (i.e., primary producers) having unlimited food resources and a detrital one in which the basal species uses detritus as a food resource. While the vast majority of theoretical studies neglects detritus, from our results it follows that the detrital food web is more stable than its grazing counterpart, because the interactions mediated by detritus damp out fluctuations in species' densities. Since the detritus model is the more complex one in terms of interaction patterns, our results provide evidence for the advocates of the complexity as one of the factors enhancing stability of ecosystems.

We study the statistical properties of spectrum and eigenstates of the Google matrix of the citation network of Physical Review for the period 1893–2009. The main fraction of complex eigenvalues with largest modulus is determined numerically by different methods based on high-precision computations with up to $p=16384$ binary digits that allow us to resolve hard numerical problems for small eigenvalues. The nearly nilpotent matrix structure allows us to obtain a semianalytical computation of eigenvalues. We find that the spectrum is characterized by the fractal Weyl law with a fractal dimension $d_f\approx 1$. It is found that the majority of eigenvectors are located in a localized phase. The statistical distribution of articles in the PageRank-CheiRank plane is established providing a better understanding of information flows on the network. The concept of ImpactRank is proposed to determine an influence domain of a given article. We also discuss the properties of random matrix models of Perron-Frobenius operators.

We present the exact rogue wave solutions of the quasi-one-dimensional inhomogeneous Gross-Pitaevskii equation by using similarity transformation. Then, by employing the exact analytical solutions we have studied the controllable behavior of rogue waves in the Bose-Einstein condensates context for the experimentally relevant systems. Additionally, we have also investigated the nonlinear tunneling of rogue waves through a conventional hyperbolic barrier and periodic barrier. We have found that, for the conventional nonlinearity barrier case, rogue waves are localized in space and time and get amplified near the barrier, while for the dispersion barrier case rogue waves are localized in space and propagating in time and their amplitude is reduced at the barrier location. In the case of the periodic barrier, the interesting dynamical features of rogue waves are obtained and analyzed analytically.

Pulsed field gradient nuclear magnetic resonance was used to determine the intrinsic longitudinal and transverse dispersivity of random packings of nearly monodisperse spheres in experiments covering 3.5 orders of magnitude in reduced velocity Pe, from the diffusion dominated regime $\mathrm{Pe}<1$ to the high velocity regime $\mathrm{Pe}>1000$. Additionally, using lattice-Boltzmann simulations with tracer tracking, the dispersivities of random packings were determined numerically. Experimental and simulation results are shown to agree to within a few percent over the full velocity range. The velocity dependence of transverse and longitudinal dispersion in packed spheres is described by heuristic models with three parameters each. The simulations were extended to regimes not accessible to experiments, dispersing tracers in flows through random “packings” of floating spheres, for comparison with results obtained using jammed packings in which spheres touch. In the regime of fast flows, the jammed packings' longitudinal dispersivities scale with hydrodynamic length. This no longer holds true in the packings with floating spheres, highlighting the role of zones of slow flow or no flow surrounding sphere-sphere contacts.