To study how nonlinear waves propagate across Y- and T-type junctions, we consider the two-dimensional (2D) sine-Gordon equation as a model and examine the crossing of kinks and breathers. Comparing energies for different geometries reveals that, for small widths, the angle of the fork plays no role. Motivated by this, we introduce a one-dimensional effective model whose solutions agree well with the 2D simulations for kink and breather solutions. These exhibit two different behaviors: a kink crosses if it has sufficient energy; conversely a breather crosses when $v>1-\omega$, where $v$ and $\omega$ are, respectively, its velocity and frequency. This methodology can be generalized to more complex nonlinear wave models.

Natural convection in a horizontal layer subject to periodic heating from above has been studied. It is shown that the primary convection leads to the cooling of the bulk of the fluid below the mean temperature of the upper wall. The secondary convection may lead either to longitudinal rolls, transverse rolls, or oblique rolls. The global flow properties (e.g., the average Nusselt number for the primary convection and the critical conditions for the secondary convection) are identical to those of the layer heated from below. However, the flow and temperature patterns exhibit phase shifts in the horizontal directions.

Using high-resolution direct numerical simulation and arguments based on the kinetic energy flux ${\Pi }_u$, we demonstrate that, for stably stratified flows, the kinetic energy spectrum $E_u\left(k\right)\sim k^{-11/5}$, the potential energy spectrum $E_{\theta }\left(k\right)\sim k^{-7/5}$, and ${\Pi }_u\left(k\right)\sim k^{-4/5}$ are consistent with the Bolgiano-Obukhov scaling. This scaling arises due to the conversion of kinetic energy to the potential energy by buoyancy. For weaker buoyancy, this conversion is weak, hence $E_u\left(k\right)$ follows Kolmogorov's spectrum with a constant energy flux. For Rayleigh-Bénard convection, we show that the energy supply rate by buoyancy is positive, which leads to an increasing ${\Pi }_u\left(k\right)$ with $k$, thus ruling out Bolgiano-Obukhov scaling for the convective turbulence. Our numerical results show that convective turbulence for unit Prandt number exhibits a constant ${\Pi }_u\left(k\right)$ and $E_u\left(k\right)\sim k^{-5/3}$ for a narrow band of wave numbers.

In this work, we analyze the phononic characteristics of wave motion in precompressed monoatomic and diatomic granular chains, with emphasis on the evolving spatial features of wave packets. A Taylor series expansion of the governing equations is considered to approximate the granular chain with the Fermi-Pasta-Ulam chain. Within this approximation, the envelope modulation of the first-order features of the wave profile is monitored and the characteristics of this modulation are determined by studying the evolution of one of the distinctive features of the spatial profile. A set of constants that describe the quantitative effects of nonlinearity on the response are determined for monoatomic and diatomic chains and interpreted as invariants of quadratic nonlinearity. The universality of these invariants is verified by constructing inverse problems to estimate the contact power law from the wave response of granular chains with arbitrary nonlinear force interaction. The imposed power law is recovered exactly from numerical simulations for a number of considered scenarios, paving the way for inverse characterization of nonlinearity from experimental data.

We report the experimental observation of the gravity-induced asymmetry for the nonlinear transformation of acoustic waves in a noncohesive granular phononic crystal. Because of the gravity, the contact precompression increases with depth inducing space variations of not only the linear and nonlinear elastic moduli but also of the acoustic wave dissipation. We show experimentally and explain theoretically that, in contrast to symmetric propagation of linear waves, the amplitude of the nonlinearly self-demodulated wave depends on whether the propagation of the waves is in the direction of the gravity or in the opposite direction. Among the observed nonlinear processes, we report frequency mixing of the two transverse-rotational modes belonging to the optical band of vibrations and propagating with negative phase velocities, which results in the excitation of a longitudinal wave belonging to the acoustic band of vibrations and propagating with positive phase velocity. We show that the measurements of the gravity-induced asymmetry in the nonlinear acoustic phenomena can be used to compare the in-depth distributions of the contact nonlinearity and of acoustic absorption.

We show that the theoretical machinery developed for the Kardar-Parisi-Zhang (KPZ) class in low dimensions is obeyed by the restricted solid-on-solid model for substrates with dimensions up to $d=6$. Analyzing different restriction conditions, we show that the height distributions of the interface are universal for all investigated dimensions. It means that fluctuations are not negligible and, consequently, the system is still below the upper critical dimension at $d=6$. The extrapolation of the data to dimensions $d\ge 7$ predicts that the upper critical dimension of the KPZ class is infinite.

The densest amorphous packing of rigid particles is known as random close packing. It has long been appreciated that higher densities are achieved by using collections of particles with a variety of sizes. For spheres, the variety of sizes is often quantified by the polydispersity of the particle size distribution: the standard deviation of the radius divided by the mean radius. Several prior studies quantified the increase of the packing density as a function of polydispersity. A particle size distribution is also characterized by its skewness, kurtosis, and higher moments, but the influence of these parameters has not been carefully quantified before. In this work, we numerically generate many sphere packings with different particle radii distributions, varying polydispersity and skewness independently of one another. We find that the packing density can increase significantly with increasing skewness and in some cases skewness can have a larger effect than polydispersity. However, the packing fraction is relatively insensitive to the higher moment value of the kurtosis. We present a simple empirical formula for the value of the random close packing density as a function of polydispersity and skewness.

The geometrical properties of the matrix blocks formed by a random fracture network are investigated numerically, for a wide range of fracture shapes and for fracture densities ranging from the dilute limit to well above the threshold where the material is entirely partitioned into finite blocks. The main block characteristics are the density and volume fraction, the mean volume and surface area, and their number of faces. In the dilute limit, general expressions for these characteristics are obtained, which provide a good approximation of the numerical data for any fracture shape. In the dense regime, most properties are governed by power laws, which involve two fitted exponents independent of the fracture shape. The shape factors identified in the dilute limit remain relevant for dense networks and can be used to formulate a general model for the block characteristics, valid up to the total matrix fracturation. The transition density when this occurs is determined. It can also be used to account for the fracture shape effects in a very simple and fairly accurate general model. Beyond the transition density, the block characteristics converge as expected toward those in the space tesselation by infinite planes.

Neural circuit motifs producing coexistent rhythmic patterns are treated as building blocks of multifunctional neuronal networks. We study the robustness of such a motif of inhibitory model neurons to reliably sustain bursting polyrhythms under random perturbations. Without noise, the exponential stability of each of the coexisting rhythms increases with strengthened synaptic coupling, thus indicating an increased robustness. Conversely, after adding noise we find that noise-induced rhythm switching intensifies if the coupling strength is increased beyond a critical value, indicating a decreased robustness. We analyze this stochastic arrhythmia and develop a generic description of its dynamic mechanism. Based on our mechanistic insight, we show how physiological parameters of neuronal dynamics and network coupling can be balanced to enhance rhythm robustness against noise. Our findings are applicable to a broad class of relaxation-oscillator networks, including Fitzhugh-Nagumo and other Hodgkin-Huxley-type networks.

Turing mechanisms can yield a large variety of patterns from noisy, homogenous initial conditions and have been proposed as patterning mechanism for many developmental processes. However, the molecular components that give rise to Turing patterns have remained elusive, and the small size of the parameter space that permits Turing patterns to emerge makes it difficult to explain how Turing patterns could evolve. We have recently shown that Turing patterns can be obtained with a single ligand if the ligand-receptor interaction is taken into account. Here we show that the general properties of ligand-receptor systems result in very large Turing spaces. Thus, the restriction of receptors to single cells, negative feedbacks, regulatory interactions among different ligand-receptor systems, and the clustering of receptors on the cell surface all greatly enlarge the Turing space. We further show that the feedbacks that occur in the FGF10-SHH network that controls lung branching morphogenesis are sufficient to result in large Turing spaces. We conclude that the cellular restriction of receptors provides a mechanism to sufficiently increase the size of the Turing space to make the evolution of Turing patterns likely. Additional feedbacks may then have further enlarged the Turing space. Given their robustness and flexibility, we propose that receptor-ligand-based Turing mechanisms present a general mechanism for patterning in biology.

The interplay of social and strategic motivations in human interactions is a largely unexplored topic in collective social phenomena. Whether individuals' decisions are taken in a purely strategic basis or due to social pressure without a rational background crucially influences the model outcome. Here we study a networked Prisoner's Dilemma in which decisions are made either based on the replication of the most successful neighbor's strategy (unconditional imitation) or by pure social imitation following an update rule inspired by the voter model. The main effects of the voter dynamics are an enhancement of the final consensus, i.e., asymptotic states are generally uniform, and a promotion of cooperation in certain regions of the parameter space as compared to the outcome of purely strategic updates. Thus, voter dynamics acts as an interface noise and has a similar effect as a pure random noise; furthermore, its influence is mostly independent of the network heterogeneity. When strategic decisions are made following other update rules such as the replicator or Moran processes, the dynamic mixed state found under unconditional imitation for some parameters disappears, but an increase of cooperation in certain parameter regions is still observed. Comparing our results with recent experiments on the Prisoner's Dilemma, we conclude that such a mixed dynamics may explain moody conditional cooperation among the agents.

We introduce a numerical method to integrate the stochastic Landau-Lifshitz-Gilbert equation in spherical coordinates for generic discretization schemes. This method conserves the magnetization modulus and ensures the approach to equilibrium under the expected conditions. We test the algorithm on a benchmark problem: the dynamics of a uniformly magnetized ellipsoid. We investigate the influence of various parameters, and in particular, we analyze the efficiency of the numerical integration, in terms of the number of steps needed to reach a chosen long time with a given accuracy.

We investigate the nonlinear dynamics of a periodic chain of bistable elements consisting of masses connected by elastic springs whose constraint arrangement gives rise to a large-deformation snap-through instability. We show that the resulting negative-stiffness effect produces three different regimes of (linear and nonlinear) wave propagation in the periodic medium, depending on the wave amplitude. At small amplitudes, linear elastic waves experience dispersion that is controllable by the geometry and by the level of precompression. At moderate to large amplitudes, solitary waves arise in the weakly and strongly nonlinear regime. For each case, we present closed-form analytical solutions and we confirm our theoretical findings by specific numerical examples. The precompression reveals a class of wave propagation for a partially positive and negative potential. The presented results highlight opportunities in the design of mechanical metamaterials based on negative-stiffness elements, which go beyond current concepts primarily based on linear elastic wave propagation. Our findings shed light on the rich effective dynamics achievable by nonlinear small-scale instabilities in solids and structures.

The deformation and breakup of Newtonian and viscoelastic droplets are studied in confined shear flow. Our numerical approach is based on a combination of lattice-Boltzmann models and finite difference schemes, the former used to model two immiscible fluids with variable viscosity ratio and the latter used to model the polymer dynamics. The kinetics of the polymers is introduced using constitutive equations for viscoelastic fluids with finitely extensible nonlinear elastic dumbbells with Peterlin's closure. We quantify the droplet response by changing the polymer relaxation time ${\tau }_P$, the maximum extensibility $L$ of the polymers, and the degree of confinement, i.e., the ratio of the droplet diameter to wall separation. In unconfined shear flow, the effects of droplet viscoelasticity on the critical capillary number ${\mathrm{Ca}}_{\mathrm{cr}}$ for breakup are moderate in all cases studied. However, in confined conditions a different behavior is observed: The critical capillary number of a viscoelastic droplet increases or decreases, depending on the maximum elongation of the polymers, the latter affecting the extensional viscosity of the polymeric solution. Force balance is monitored in the numerical simulations to validate the physical picture.

In the framework of a two-defect model, based on a variation of the Landau technique, the kinetics of structural defect generation in solids under severe external load is investigated. The approach is based on a special form of kinetic equation in terms of internal energy, which is applied here to the description of an important practical problem of fine-grained structure formation in metals under severe plastic deformation. It unifies strengthening curves over the entire range of deformation, including the Hall-Petch and linear strengthening sections.

We studied how to obtain a distribution for the number of ancestors in species of sexual reproduction. Present models concentrate on the estimation of distributions repetitions of ancestors in genealogical trees. It has been shown that it is not possible to reconstruct the genealogical history of each species along all its generations by means of a geometric progression. This analysis demonstrates that it is possible to rebuild the tree of progenitors by modeling the problem with a Markov chain. For each generation, the maximum number of possible ancestors is different. This presents huge problems for the resolution. We found a solution through a dilation of the sample space, although the distribution defined there takes smaller values with respect to the initial problem. In order to correct the distribution for each generation, we introduced the invariance under a gauge (local) group of dilations. These ideas can be used to study the interaction of several processes and provide a new approach on the problem of the common ancestor. In the same direction, this model also provides some elements that can be used to improve models of animal reproduction.

We report in this paper a theoretical study on the quantum thermodynamic properties of a fractional damping system. Through the analysis, few nontrivial characteristics are revealed, which include (1) a fractional power-law decay entropy function, which provides an evidence for the validity of the third law of thermodynamics in the quantum dissipative region and (2) the varying of the entropy from a nonlinear divergent function to a semilinear decay function with a fractional exponent as the temperature approaches absolute zero.

One of the interesting phenomena due to topological heterogeneities in complex networks is the friendship paradox: Your friends have on average more friends than you do. Recently, this paradox has been generalized for arbitrary node attributes, called the generalized friendship paradox (GFP). The origin of GFP at the network level has been shown to be rooted in positive correlations between degrees and attributes. However, how the GFP holds for individual nodes needs to be understood in more detail. For this, we first analyze a solvable model to characterize the paradox holding probability of nodes for the uncorrelated case. Then we numerically study the correlated model of networks with tunable degree-degree and degree-attribute correlations. In contrast to the network level, we find at the individual level that the relevance of degree-attribute correlation to the paradox holding probability may depend on whether the network is assortative or dissortative. These findings help us to understand the interplay between topological structure and node attributes in complex networks.

We study inertial-range dynamics and scaling laws in unforced two-dimensional magnetohydrodynamic turbulence in the regime of moderately small and small initial magnetic-to-kinetic-energy ratio $r_0$, with an emphasis on the latter. The regime of small $r_0$ corresponds to a relatively weak field and strong magnetic stretching, whereby the turbulence is characterized by an intense conversion of kinetic into magnetic energy (dynamo action in the three-dimensional context). This conversion is an inertial-range phenomenon and, upon becoming quasisaturated, deposits the converted energy within the inertial range rather than transferring it to the small scales. As a result, the magnetic-energy spectrum $E_{\mathit{b}}\left(k\right)$ in the inertial range can become quite shallow and may not be adequately explained or understood in terms of conventional cascade theories. It is demonstrated by numerical simulations at high Reynolds numbers (and unity magnetic Prandtl number) that the energetics and inertial-range scaling depend strongly on $r_0$. In particular, for fully developed turbulence with $r_0$ in the range $[1/4,1/4096]$, $E_{\mathit{b}}\left(k\right)$ is found to scale as $k^{\alpha }$, where $\alpha \gtrsim -1$, including $\alpha >0$. The extent of such a shallow spectrum is limited, becoming broader as $r_0$ is decreased. The slope $\alpha$ increases as $r_0$ is decreased, appearing to tend to $+1$ in the limit of small $r_0$. This implies equipartition of magnetic energy among the Fourier modes of the inertial range and the scaling $k^{-1}$ of the magnetic potential variance, whose flux is direct rather than inverse. This behavior of the potential resembles that of a passive scalar. However, unlike a passive scalar whose variance dissipation rate slowly vanishes in the diffusionless limit, the dissipation rate of the magnetic potential variance scales linearly with the diffusivity in that limit. Meanwhile, the kinetic-energy spectrum is relatively steep, followed by a much shallower tail due to strong antidynamo excitation. This gives rise to a total-energy spectrum poorly obeying a power-law scaling.

We investigate the scaling behavior of sample statistics of pore-scale Lagrangian velocities in two different rock samples, Bentheimer sandstone and Estaillades limestone. The samples are imaged using x-ray computer tomography with micron-scale resolution. The scaling analysis relies on the study of the way $q\text{th}$-order sample structure functions (statistical moments of order $q$ of absolute increments) of Lagrangian velocities depend on separation distances, or lags, traveled along the mean flow direction. In the sandstone block, sample structure functions of all orders exhibit a power-law scaling within a clearly identifiable intermediate range of lags. Sample structure functions associated with the limestone block display two diverse power-law regimes, which we infer to be related to two overlapping spatially correlated structures. In both rocks and for all orders $q$, we observe linear relationships between logarithmic structure functions of successive orders at all lags (a phenomenon that is typically known as extended power scaling, or extended self-similarity). The scaling behavior of Lagrangian velocities is compared with the one exhibited by porosity and specific surface area, which constitute two key pore-scale geometric observables. The statistical scaling of the local velocity field reflects the behavior of these geometric observables, with the occurrence of power-law-scaling regimes within the same range of lags for sample structure functions of Lagrangian velocity, porosity, and specific surface area.