Soliton and breather solutions of the nonlinear Schr\"odinger equation (NLSE) are known to model localized structures in nonlinear dispersive media such as on the water surface. One of the conditions for an accurate propagation of such exact solutions is the proper generation of the exact initial phase-shift profile in the carrier wave, as defined by the NLSE envelope at a specific time or location. Here, we show experimentally the significance of such initial exact phase excitation during the hydrodynamic propagation of localized envelope solitons and breathers, which modulate a plane wave of constant amplitude (finite background). Using the example of stationary black solitons in intermediate water depth and pulsating Peregrine breathers in deep-water, we show how these localized envelopes disintegrate while they evolve over a long propagation distance when the initial phase shift is zero. By setting the envelope phases to zero, the dark solitons will disintegrate into two gray-type solitons and dispersive elements. In the case of the doubly-localized Peregrine breather the maximal amplification is considerably retarded; however locally, the shape of the maximal focused wave measured together with the respective signature phase-shift are almost identical to the exact analytical Peregrine characterization at its maximal compression location. The experiments, conducted in two large-scale shallow-water as well as deep-water wave facilities, are in very good agreement with NLSE simulations for all cases.

The Painlev\'e property for a (2+1)-dimensional Korteweg-de Vries (KdV) extension, the combined KP3 (Kadomtsev- Petviashvili) and KP4 (cKP3-4) is proved by using Kruskal's simplification. The truncated Painlev\'e expansion is used to find the Schwartz form, the B\"acklund/Levi transformations and the residual nonlocal symmetry. The residual symmetry is localized to find its finite B\"acklund transformation. The local point symmetries of the model constitute a centerless Kac-Moody-Virasoro algebra. The local point symmetries are used to find the related group invariant reductions including a new Lax integrable model with a fourth order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.

In this paper, a methodology inspired on bond and site percolation methods is applied to the estimation of the resilience against failures in power grids. Our approach includes vulnerability measures with both dynamical and structural foundations as an attempt to find more insights about the relationships between topology and dynamics in the second-order Kuramoto model on complex networks. As test cases for numerical simulations, we use the real-world topology of the Colombian power transmission system, as well as randomly generated networks with spatial embedding. It is observed that, by focusing the attacks on those dynamical vulnerabilities, the power grid becomes, in general, more prone to reach a state of total blackout, which in the case of node removal procedures it is conditioned by the homogeneity of power distribution in the network.

The Kuramoto model, which serves as a paradigm for investigating synchronization phenomenon of oscillatory system, is known to exhibit second-order, i.e., continuous, phase transitions in the macroscopic order parameter. Here, we generalize a number of classical results by presenting a general framework for capturing, analytically, the critical scaling of the order parameter at the onset of synchronization. Using a self-consistent approach and constructing a characteristic function, we identify various phase transitions toward synchrony and establish scaling relations describing the asymptotic dependence of the order parameter on coupling strength near the critical point. We find that the geometric properties of the characteristic function, which depends on the natural frequency distribution, determines the scaling properties of order parameter above the criticality.

Dynamical systems with long delay feedback can exhibit complicated temporal phenomena, which once re-organized in a two-dimensional space are reminiscent of spatio-temporal behavior. In this framework, normal forms description have been developed to reproduce the dynamics and the opportunity to treat the corresponding variables as true space and time has been since established. However, recently an alternative approach has been proposed in Ref. \cite{Marino2018} with a different interpretation of the variables involved, which takes better into account their physical character and allows for an easier determination of the normal forms. In this paper, we extend such idea and apply it to a number of paradigmatic examples, paving the way to a re-thinking of the concept of spatio-temporal representation of long-delayed systems.

Game of Life is a simple and elegant model to study dynamical system over networks. The model consists of a graph where every vertex has one of two types, namely, dead or alive. A configuration is a mapping of the vertices to the types. An update rule describes how the type of a vertex is updated given the types of its neighbors. In every round, all vertices are updated synchronously, which leads to a configuration update. While in general, Game of Life allows a broad range of update rules, we focus on two simple families of update rules, namely, underpopulation and overpopulation, that model several interesting dynamics studied in the literature. In both settings, a dead vertex requires at least a desired number of live neighbors to become alive. For underpopulation (resp., overpopulation), a live vertex requires at least (resp. at most) a desired number of live neighbors to remain alive. We study the basic computation problems, e.g., configuration reachability, for these two families of rules. For underpopulation rules, we show that these problems can be solved in polynomial time, whereas for overpopulation rules they are PSPACE-complete.

In linear science, the wave motion equation with general D'Alembert wave solutions is one of the fundamental models. The D'Alembert wave is an arbitrary travelling wave moving along one direction under a fixed model (material) dependent velocity. However, the D'Alembert waves are missed when nonlinear effects are introduced to wave motions. In this paper, we study the possible travelling wave solutions, multiple soliton solutions and soliton molecules for a special (2+1)-dimensional Koteweg-de Vries (KdV) equation, the so-called Nizhnik-Novikov-Veselov (NNV) equation. The missed D'Alembert wave is re-discovered from the NNV equation. By using the velocity resonance mechanism, the soliton molecules are found to be closely related to D'Alembert waves. In fact, the soliton molecules of the NNV equation can be viewed as special D'Alembert waves. The interaction solutions among special D'Alembert type waves ($n$-soliton molecules and soliton-solitoff molecules) and solitons are also discussed.

In this paper, a susceptible-infected-removed (SIR) model has been used to track the evolution of the spread of the COVID-19 virus in four countries of interest. In particular, the epidemic model, that depends on some basic characteristics, has been applied to model the time evolution of the disease in Italy, India, South Korea and Iran. The economic, social and health consequences of the spread of the virus have been cataclysmic. Hence, it is essential that available mathematical models can be developed and used for the comparison to be made between published data sets and model predictions. The predictions estimated from the SIR model here, can be used in both the qualitative and quantitative analysis of the spread. It gives an insight into the spread of the virus that the published data alone cannot do by updating them and the model on a daily basis. For example, it is possible to detect the early onset of a spike in infections or the development of a second wave using our modeling approach. We considered data from March to June, 2020, when different communities are severely affected. We demonstrate predictions depending on the model's parameters related to the spread of COVID-19 until September 2020. By comparing the published data and model results, we conclude that in this way, it may be possible to better reflect the success or failure of the adequate measures implemented by governments and individuals to mitigate and control the current pandemic.

Numerical modeling is used to investigate the dynamics of a polaron in a chain with small random Langevin-like perturbations which imitate the environmental temperature $T$ and under the influence of a constant electric field. In the semiclassical Holstein model the region of existence of polarons in the thermodynamic equilibrium state depends not only on temperature but also on the chain length. Therefore when we compute dynamics from initial polaron data, the mean displacement of the charge mass center differs for different-length chains at the same temperature. For a large radius polaron, it is shown numerically that the ``mean polaron displacement'' (which takes account only of the polaron peak and its position) behaves similarly for different-length chains during the time when the polaron persists. A similar slope of the polaron displacement enables one to find the polaron mean velocity and, by analogy with the charge mobility, assess the ``polaron mobility''. The calculated values of the polaron mobility for $T \approx 0$ are close to the value at $T=0$, which is small but not zero. For the parameters corresponding to the small radius polaron, simulations of dynamics demonstrate switching mode between immobile polaron and delocalized state. The position of the new polaron is not related to the position of the previous one; charge transfer occurs in the delocalized state.

The problem of data-driven identification of coherent observables of measure-preserving, ergodic dynamical systems is studied using kernel integral operator techniques. An approach is proposed whereby complex-valued observables with approximately cyclical behavior are constructed from a pair eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are $\epsilon$-approximate eigenfunctions of the Koopman evolution operator of the system, with a bound $\epsilon$ controlled by the length of the delay-embedding window, the evolution time, and appropriate spectral gap parameters. In particular, $ \epsilon$ can be made arbitrarily small as the embedding window increases so long as the corresponding eigenvalues remain sufficiently isolated in the spectrum of the integral operator. It is also shown that the time-autocorrelation functions of such observables are $\epsilon$-approximate Koopman eigenvalue, exhibiting a well-defined characteristic oscillatory frequency (estimated using the Koopman generator) and a slowly-decaying modulating envelope. The results hold for measure-preserving, ergodic dynamical systems of arbitrary spectral character, including mixing systems with continuous spectrum and no non-constant Koopman eigenfunctions in $L^2$. Numerical examples reveal a coherent observable of the Lorenz 63 system whose autocorrelation function remains above 0.5 in modulus over approximately 10 Lyapunov timescales.

Dissipating of disorder quantum vortices in an annular two-dimensional Bose-Einstein condensate can form a macroscopic persistent flow of atoms. We propose a protocol to create persistent flow with high winding number based on a double concentric ring-shaped configuration. We find that a sudden geometric quench of the trap from single ring-shape into double concentric ring-shape will enhance the circulation flow in the outer ring-shaped region of the trap when the initial state of the condensate is with randomly distributed vortices of the same charge. The circulation flows that we created are with high stability and good uniformity free from topological excitations. Our study is promising for new atomtronic designing, and is also helpful for quantitatively understanding quantum tunneling and interacting quantum systems driven far from equilibrium.

This article provides a focused review of recent findings which demonstrate, in some cases quite counter-intuitively, the existence of bound states with a singularity of the density pattern at the center, while the states are physically meaningful because their total norm converges. One model of this type is based on the 2D Gross-Pitaevskii equation (GPE) which combines the attractive potential ~ 1/r^2 and the quartic self-repulsive nonlinearity, induced by the Lee-Huang-Yang effect (quantum fluctuations around the mean-field state). The GPE demonstrates suppression of the 2D quantum collapse, driven by the attractive potential, and emergence of a stable ground state (GS), whose density features an integrable singularity ~1/r^{4/3} at r --> 0. Modes with embedded angular momentum exist too, and they have their stability regions. A counter-intuitive peculiarity of the model is that the GS exists even if the sign of the potential is reversed from attraction to repulsion, provided that its strength is small enough. This peculiarity finds a relevant explanation. The other model outlined in the review includes 1D, 2D, and 3D GPEs, with the septimal (seventh-order), quintic, and cubic self-repulsive terms, respectively. These equations give rise to stable singular solitons, which represent the GS for each dimension D, with the density singularity ~1/r^{2/(4-D). Such states may be considered as a result of screening of a "bare" delta-functional attractive potential by the respective nonlinearity.

Neurodegenerative diseases are closely associated with the amplification and invasion of toxic proteins. In particular Alzheimer's disease is characterized by the systematic progression of amyloid-$\beta$ and $\tau$-proteins in the brain. These two protein families are coupled and it is believed that their joint presence greatly enhances the resulting damage. Here, we examine a class of coupled chemical kinetics models of healthy and toxic proteins in two spatial dimensions. The anisotropic diffusion expected to take place within the brain along axonal pathways is factored in the models and produces a filamentary, predominantly one-dimensional transmission. Nevertheless, the potential of the anisotropic models towards generating interactions taking advantage of the two-dimensional landscape is showcased. Finally, a reduction of the models into a simpler family of generalized Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) type systems is examined. It is seen that the latter captures well the qualitative propagation features, although it may somewhat underestimate the concentrations of the toxic proteins.

The ExB electron drift instability, present in many plasma devices, is an important agent in cross-field particle transport. In presence of a resulting low frequency electrostatic wave, the motion of a charged particle becomes chaotic and generates a stochastic web in phase space. We define a scaling exponent to characterise transport in phase space and we show that the transport is anomalous, of super-diffusive type. Given the values of the model parameters, the trajectories stick to different kinds of islands in phase space, and their different sticking time power-law statistics generate successive regimes of the super-diffusive transport.

We show that turbulent dynamics that arise in simulations of the three-dimensional Navier--Stokes equations in a triply-periodic domain under sinusoidal forcing can be described as transient visits to the neighborhoods of unstable time-periodic solutions. Based on this description, we reduce the system with more than $10^5$ degrees of freedom to an 18-node Markov chain where each node corresponds to the neighborhood of a periodic orbit. The model accurately reproduces long-term averages of the system's observables as weighted sums over the periodic orbits.

We derive the equations of motion for the dynamics of a porous media filled with an incompressible fluid. We use a variational approach with a Lagrangian written as the sum of terms representing the kinetic and potential energy of the elastic matrix, and the kinetic energy of the fluid, coupled through the constraint of incompressibility. As an illustration of the method, the equations of motion for both the elastic matrix and the fluid are derived in the spatial (Eulerian) frame. Such an approach is of relevance e.g. for biological problems, such as sponges in water, where the elastic porous media is highly flexible and the motion of the fluid has a 'primary' role in the motion of the whole system. We then analyze the linearized equations of motion describing the propagation of waves through the media. In particular, we derive the propagation of S-waves and P-waves in an isotropic media. We also analyze the stability criteria for the wave equations and show that they are equivalent to the physicality conditions of the elastic matrix. Finally, we show that the celebrated Biot's equations for waves in porous media are obtained for certain values of parameters in our models.

Many physical systems display quantized energy states. In optics, interacting resonant cavities show a transmission spectrum with split eigenfrequencies, similar to the split energy levels that result from interacting states in bonded multi-atomic, i.e. molecular, systems. Here, we study the nonlinear dynamics of photonic diatomic molecules in linearly coupled microresonators and demonstrate that the system supports the formation of self-enforcing solitary waves when a laser is tuned across a split energy level. The output corresponds to a frequency comb (microcomb) whose characteristics in terms of power spectral distribution are unattainable in single-mode (atomic) systems. Photonic molecule microcombs are coherent, reproducible, and reach high conversion efficiency and spectral flatness whilst operated with a laser power of a few milliwatts. These properties can favor the heterogeneous integration of microcombs with semiconductor laser technology and facilitate applications in optical communications, spectroscopy and astronomy.

Several experiments have demonstrated the existence of an electro-mechanical effect in many biological tissues and hydrogels, and its actual influence on growth, migration, and pattern formation. Here, to model these interactions and capture some growth phenomena found in Nature, we extend volume growth theory to account for an electro-elasticity coupling. Based on the multiplicative decomposition, we present a general analysis of isotropic growth and pattern formation of electro-elastic solids under external mechanical and electrical fields. As an example, we treat the case of a tubular structure to illustrate an electro-mechanically guided growth affected by axial strain and radial voltage. Our numerical results show that a high voltage can enhance the non-uniformity of the residual stress distribution and induce extensional buckling, while a low voltage can delay the onset of wrinkling shapes and can also generate more complex morphologies. Within a controllable range, axial tensile stretching shows the ability to stabilise the tube and help form more complex 3D patterns, while compressive stretching promotes instability. Both the applied voltage and external axial strain have a significant impact on guiding growth and pattern formation. Our modelling provides a basic tool for analysing the growth of electro-elastic materials, which can be useful for designing a pattern prescription strategy or growth self-assembly in Engineering.

Originally arising in the context of interacting particle systems in statistical physics, dynamical systems and differential equations on networks/graphs have permeated into a broad number of mathematical areas as well as into many applications. One central problem in the field is to find suitable approximations of the dynamics as the number of nodes/vertices tends to infinity, i.e., in the large graph limit. A cornerstone in this context are Vlasov-Fokker-Planck equations (VFPEs) describing a particle density on a mean-field level. For all-to-all coupled systems, it is quite classical to prove the rigorous approximation by VFPEs for many classes of particle systems. For dense graphs converging to graphon limits, one also knows that mean-field approximation holds for certain classes of models, e.g., for the Kuramoto model on graphs. Yet, the space of intermediate density and sparse graphs is clearly extremely relevant. Here we prove that the Kuramoto model can be be approximated in the mean-field limit by far more general graph limits than graphons. In particular, our contributions are as follows. (I) We show, how to introduce operator theory more abstractly into VFPEs by considering graphops. Graphops have recently been proposed as a unifying approach to graph limit theory, and here we show that they can be used for differential equations on graphs. (II) For the Kuramoto model on graphs we rigorously prove that there is a VFPE equation approximating it in the mean-field sense. (III) This mean-field VFPE involves a graphop, and we prove the existence, uniqueness, and continuous graphop-dependence of weak solutions. (IV) On a technical level, our results rely on designing a new suitable metric of graphop convergence and on employing Fourier analysis on compact abelian groups to approximate graphops using summability kernels.