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In the present work we explore the path from a harmonic to a biharmonic PDE of Klein-Gordon type from a continuation/bifurcation perspective. More specifically, we make use of the Riesz fractional derivative as a tool that allows us to interpolate between these two limits. We illustrate, in particular, how the coherent kink structures existing in these models transition from the exponential tail of the harmonic operator case, via the power-law tails of intermediate fractional orders, to the oscillatory exponential tails of the biharmonic model. Importantly, we do not limit our considerations to the single kink case, but extend to the kink-antikink pair, finding an intriguing cascade of saddle-center bifurcations happening exponentially close to the biharmonic limit. Our analysis clearly explains the transition between the infinitely many stationary soliton pairs of the biharmonic case and the absence of even a single such pair in the harmonic limit. The stability of the different configurations obtained and the associated dynamics and phase portraits are also analyzed.

We study quasi-periodic eigenvalue problems that arise in the stability analysis of periodic traveling wave solutions to Hamiltonian PDEs. We establish bounds on regions in the complex plane when the eigenvalues may deviate from the imaginary axis, and estimates for the number of such off-axis eigenvalues. These relations hold when the dispersion relation grows sufficiently rapidly in the wavenumber. The proofs involve a Gershgorin disk argument together with the Hamiltonian symmetry of the spectrum. The results are applicable to a broad class of nonlinear dispersive equations including the generalized Korteweg--de Vries, Benjamin--Bona--Mahoney, and Kawanhara equations., Comment: 20 pages; 3 figures

This paper addresses the problem of parallelizing computations to study non-linear dynamics in large networks of non-locally coupled oscillators using heterogeneous computing resources. The proposed approach can be applied to a variety of non-linear dynamics models with runtime specification of parameters and network topologies. Parallelizing the solution of equations for different network elements is performed transparently and, in contrast to available tools, does not require parallel programming from end-users. The runtime scheduler takes into account the performance of computing and communication resources to reduce downtime and to achieve a quasi-optimal parallelizing speed-up. The proposed approach was implemented, and its efficiency is proven by numerous applications for simulating large dynamical networks with 10^3-10^8 elements described by Hodgkin-Huxley, FitzHugh-Nagumo, and Kuramoto models, for investigating pathological synchronization during Parkinson's disease, analyzing multi-stability, for studying chimera and solitary states in 3D networks, etc. All the above computations may be performed using symmetrical multiprocessors, graphic processing units, and a network of workstations within the same run and it was demonstrated that near-linear speed-up can be achieved for large networks. The proposed approach is promising for extension to new hardware like edge-computing devices., Comment: 12 pages, 11 figures

We investigate the interaction between two flat-top solitons within the cubic-quintic nonlinear Schr\"odinger equation framework. Our study results point towards a significant departure of flat-top solitons collisional characteristics from the conventional behaviors exhibited in the scattering dynamics of two bright solitons. Our investigation outlines specific regimes corresponding to the dual flat-top solitons' elastic and inelastic collisions. We determine regimes in the parameter space where exchange in the widths of the interacting flat-top solitons is possible, even within the elastic collision domain. We find a periodic occurrence of completely elastic scattering of flat-top solitons in terms of the solitons' parameters. Investigating the internal energy transfers between the solitons and the emitted radiation reveals the origin of inelasticity in flat-top solitons collisions. We perform a variational calculation that accounts for the amount of radiation produced by the collision and hence provides further insight on the physics underlying the loss of elasticity of collisions., Comment: 21 pages, 19 figures

A simple model and computation of Moir\'e-regulated composition evolution kinetics of bicomponent nanoclusters is presented. Assuming continuous adsorbate coverage on top of 2D bilayer and Moir\'e potential-driven nanocluster formation at fcc sites of Moir\'e landscape, these sites experience the influx of one component of a bicomponent adsorbate and the outflux of another component. Kinetics of this process is characterized for several combinations of adsorption potentials and their relative strengths.

The large-scale dynamics of heat spots in a thin layer of incompressible rotating fluid under the action of Coriolis and gravity is considered to obtain a simple model of synoptic turbulence. The derivation of equations describing the evolution of the spots is based on the use of the variational principle, perturbation theory, and the assumption of geostrophic balance. Exact solutions with piecewise constant contour curvature are found. The simplest of them look like circular spots that move with constant velocity depending on their radius. More complex solutions, such as ``loop solitons'', are shown to have fractal structure and are constructed by the gluing method. Heat spots can act as structural elements of turbulence. In particular, we show that the spectral energy density for the velocity field of a random ensemble of heat spots has the power asymptotics $E\propto k^{\alpha}$ with exponents $\alpha=2$ at $k\bar{R}<1$, and $\alpha=-1$ at $k\bar{R}>1$, where $\bar{R}$ is the average (over ensemble) radius of spot., Comment: 5 pages, 2 figures

Multiple stationary, localized structures were recently found for inclined porous medium convection with constant-temperature boundaries. We analyze traveling asymmetric, localized convective structures, consisting of 1 to 5 pulses, in a 2D inclined porous layer with fixed temperature at the bottom and an imperfectly conducting boundary at the top, such that midplane reflection symmetry is broken. Direct numerical simulations (DNS) are performed with different Biot numbers at the top boundary. The drift velocity $c$ of pulses is measured for different values of the symmetry parameter $\kappa\geq0$ based on the Biot number, with perfect midplane reflection symmetry and $c=0$ at $\kappa=0$. In small domains, the drift velocity $c>0$ (upslope), increases monotonically with $\kappa$, while in large domains $c$ changes sign depending on parameters. We show that pulse tails, controlling interactions, depend on the dominant spatial eigenvalues, whose real part is closest to zero, with a transition at $\kappa_c>0$: below $\kappa_c$, both dominant eigenvalues are complex, and the tails are oscillatory. Above $\kappa_c$, the dominant spatial eigenvalue with a positive real part becomes real, and the downslope tail transitions from oscillatory to monotonic. Below $\kappa_c$, bound states with different numbers of pulses exist whose collisions are studied. Well above $\kappa_c$, adjacent pulses repel each other, spreading out and becoming equispaced in the domain. A reduced model is proposed based on the interaction via tails of adjacent pulses, reproducing the repulsion and collisions from DNS. The model shows that the transition from bound states to equidistant spreading occurs when the monotonic/oscillatory tails have the same slope. This study elucidates the motion of localized patterns in moderate-Rayleigh number convection in an inclined porous layer with an imperfectly conducting boundary.

In addition to a common synchronization and/or localization behavior, a system of linearly coupled identical bistable Van der Pol (BVdP) oscillators can exhibit a "non-conventional" or "modal" synchronization. In two-DOF case, one can observe stable beatings attractor with synchronized amplitudes of the symmetric and antisymmetric modes. Current study demonstrates that this unusual behavior is generic and quite ubiquitous, if the system is explored in appropriate parametric regime. Indeed, in the absence of the self - excitation the system of coupled linear oscillators possesses a complete set of non-interacting eigenmodes. If the system is symmetric and the coupling is weak enough, appropriate initial conditions will result in a continuous multi-parametric family of stationary beatings or beat waves. If the self-excitation terms are small enough, i.e. even weaker than the weak coupling, one can expect that, generically, some special stationary or wave beatings will turn into the stable attractors. We demonstrate this phenomenon in systems of ring-coupled BVdP oscillators. In particular, we observe simple two-wave synchronization for $N=2,3,5,6,7$ coupled oscillators; for $N=2$ it corresponds to the "modal" synchronization observed previously. The case $N=4$ is special: due to internal resonances in the slow flow, the two-wave synchronization turns unstable, and more complicated patterns of the multi-wave synchronization are revealed. Analytic models manage to capture the shapes of the observed beat waves. All results are verified by direct numeric simulations.

In this paper the finite-difference time-domain general vector auxiliary differential equation method [Greene, J. H. and A. Taflove, Opt. Express 14, 8305 (2006)], nonlinear polarization vector is extended to include anisotropy, in nonlinear isotropic media at optical frequencies. The theory is presented for extending the method in 3D Cartesian coordinates, and then simulation results are presented for the simplified 2D transverse magnetic polarization case, revisiting the fused silica example introduced in the 2006 paper including the anisotropic part of the nonlinear polarization vector. The simulation results including the anisotropic part of the polarization vector were compared with isotropic polarization vector simulation results., Comment: 15 pages, 2 figures

We study transmission stabilization of optical solitons against emission of radiation in nonlinear optical waveguides in the presence of weak linear gain-loss, cubic loss, and the collisional Raman frequency shift. We first show how the collisional Raman frequency shift perturbation arises in three different physical setups. We then show by numerical simulations with a perturbed nonlinear Schr\"odinger (NLS) model that transmission in waveguides with weak frequency-independent linear gain is unstable. The radiative instability is stronger than the radiative instabilities that were observed in earlier studies for soliton transmission in the presence of weak linear gain, cubic loss, and various frequency-shifting physical mechanisms. In particular, the Fourier spectrum of the radiation is significantly more spiky and broadband than the radiation's Fourier spectra in earlier studies. Moreover, we demonstrate by numerical simulations with another perturbed NLS model that transmission in waveguides with weak frequency-dependent linear gain-loss, cubic loss, and the collisional Raman frequency shift is stable. Despite the stronger radiative instability in the corresponding waveguide setup with weak linear gain, stabilization occurs via the same generic mechanism that was suggested in earlier studies. More precisely, the collisional Raman frequency shift experienced by the soliton leads to the separation of the soliton's and the radiation's Fourier spectra, while the frequency-dependent linear gain-loss leads to efficient suppression of radiation emission. Thus, our study demonstrates the robustness of the proposed generic soliton stabilization method, which is based on the interplay between perturbation-induced shifting of the soliton's frequency and frequency-dependent linear gain-loss., Comment: The paper demonstrates stabilization of optical solitons against radiation emission in the presence of weak linear gain-loss, cubic loss, and the collisional Raman frequency shift. It significantly extends the stabilization method that was proposed in arXiv:1804.03226 by showing that the proposed method works even when the underlying radiative instability due to linear gain is strong

In this work, we study the asymptotic behaviors and dynamics of degenerate and mixed solitons for the coupled Hirota system with strong coherent coupling effects in the isotropic nonlinear medium. Using the binary Darboux transformation, we derive the solutions to represent the degenerate solitons with two eigenvalues that are conjugate to each other. We obtain three types of degenerate solitons and provide their asymptotic expressions. Notably, these degenerate solitons exhibit time-dependent velocities, and the relative distance between the two asymptotic solitons increases logarithmically with the higher-order perturbation parameter $|\varepsilon|$ increasing. We also asymptotically reveal four interaction mechanisms between a degenerate soliton and a bell-shaped soliton: (1) elastic interaction with a position shift; (2) inelastic interaction for the degenerate soliton but elastic for the bell-shaped one; (3) elastic interaction for the degenerate soliton but inelastic for the bell-shaped one; and (4) the coherent interaction during a longer interaction region and elastic interaction based on specific parameter conditions. Besides, we analyze a special degenerate vector soliton that exhibits significant coherence effects, and numerically study the relationship between the robustness of such solitons and parameter $\varepsilon$. Our results indicate that $\varepsilon$ significantly affects the coherence of these solitons, and their robustness decreases when $|\varepsilon|$ increases.

The currents in the ocean have a serious impact on ocean dynamics, since they affect the transport of mass and thus the distribution of salinity, nutrients and pollutants. In many physically important situations the current depends quadratic-ally on the depth. We consider a single layer of fluid and study the propagation of the surface waves in the presence of depth-dependent current with quadratic profile. We select the scale of parameters and quantities, which are typical for the Boussinesq propagation regime (long wave and small amplitude limit) and we also derive the well known KdV model for the surface waves interacting with current., Comment: 16 pages, 1 figure

In the present work, we study the nonlinear dynamics of a microtubule, an important part of the cytoskeleton. We use a two-component model of the relevant system. A crucial nonlinear differential equation is solved with semi-discrete approximation, yielding some localised modulated solitary waves called the breathers. A detailed estimation of the existing parameters is provided. The numerical investigation shows that the solutions are robust only if the carrier velocity of the breather wave is higher than its envelope velocity. That disproves the previously accepted solutions based on the equality of these velocities.

Networks of nonlocally coupled leaky Integrate-and-Fire neurons exhibit a variety of complex collective behaviors, such as partial synchronization, frequency or amplitude chimeras, solitary states and bump states. In particular, the bump states consist of one or many regions of asynchronous elements within a sea of quiescent subthreshold elements. The asynchronous domains (also called bumps) travel in the network in a direction predetermined by the initial conditions. In the present study we investigate the occurrence of bump states in networks of leaky Integrate-and-Fire neurons in two-dimensions using nonlocal toroidal connectivity and we explore possible mechanisms for stabilizing the moving asynchronous domains. Our findings indicate that I) incorporating a refractory period can effectively anchor the position of these domains in the network, and II) the switching off of some randomly preselected nodes (i.e., making them permanently idle/inactive) can likewise contribute to stabilizing the positions of the asynchronous domains. In particular, in case II for large values of the coupling strength and a large percentage of idle elements, all nodes acquire different fixed (frozen) values in the subthreshold region and oscillations cease throughout the network due to self-organization. For the special case of stationary bump states, we propose an analytical approach to predict their properties. This approach is based on the self-consistency argument and is valid for infinitely large networks. Case I is of particular biomedical interest in view of the importance of refractoriness for biological neurons, while case II can be biomedically relevant when designing therapeutic methods for stabilizing moving signals in the brain.

We derive a simple model of actin waves consisting of three partial differential equations (PDEs) for active and inactive GTPase promoting growth of filamentous actin (F-actin, $F$). The F-actin feeds back to inactivate the GTPase at rate $sF$, where $s\ge 0$ is a ``negative feedback'' parameter. In contrast to previous models for actin waves, the simplicity of this model and its geometry (1D periodic cell perimeter) permits a full PDE bifurcation analysis. Based on a combination of continuation methods, linear stability analysis, and PDE simulations, we explore the existence, stability, interactions, and transitions between homogeneous steady states (resting cells), wave-pinning (polar cells), and travelling waves (cells with ruffling protrusions). We show that the value of $s$ and the size of the cell (length of perimeter) can affect the existence, coexistence, and stability of the patterns, as well as dominance of the one or another cell state. Implications to motile cells are discussed., Comment: 25 pages, 13 figures

In the past decade, photo-induced superconducting-like behaviors have been reported in a number of materials driven by intense pump fields. Of particular interest is the high-$T_c$ cuprate $\rm Y Ba_2 Cu_2 O_{6+x}$, where such effect has been reported up to the so-called pseudogap temperature $T^* \sim 300-400$ K. In a recent tour-de-force experiment, a transient magnetic field which is proportional to and in the same direction of an applied field has been observed outside the sample, suggestive of flux exclusion due to the Meissner effect. In this paper, we point out that the transient magnetic field could be explained by a model of bilayers of copper-oxygen planes with a local superconducting phase variable persisting up to the pseudo-gap temperature at equilibrium. Under pumping, the time evolution is described by a driven sine-Gordon equation. In the presence of an external magnetic field, this model exhibits a novel instability which amplifies the current at the edges of the bilayer formed by defects or grain boundaries, producing a giant paramagnetic magnetization in the same direction as the applied field. We present how this scenario can fit most of the available data and propose additional experimental tests which can distinguish our proposal from the Meissner flux exclusion scenario. To the extent that this model can account for the data, we conclude that the experiments have the important consequence of revealing the presence of local pairing in the pseudogap phase. More broadly, this work provides a new mechanism for amplifying external magnetic fields at ultra-fast time scales., Comment: 11 pages, 5 figures + Supplementary

We introduce a waveguiding system composed of three linearly-coupled fractional waveguides, with a triangular (prismatic) transverse structure. It may be realized as a tri-core nonlinear optical fiber with fractional group-velocity dispersion (GVD), or, possibly, as a system of coupled Gross--Pitaevskii equations for a set of three tunnel-coupled cigar-shaped traps filled by a Bose-Einstein condensate of particles moving by L\'evy flights. The analysis is focused on the phenomenon of spontaneous symmetry breaking (SSB) between components of triple solitons, and the formation and stability of vortex modes. In the self-focusing regime, we identify symmetric and asymmetric soliton states, whose structure and stability are determined by the L\'evy index of the fractional GVD, the inter-core coupling strength, and the total energy, which determines the system's nonlinearity. Bifurcation diagrams (of the supercritical type) reveal regions where SSB occurs, identifying the respective symmetric and asymmetric ground-state soliton modes. In agreement with the general principle of the SSB theory, the solitons with broken inter-component symmetry prevail with the increase of the energy in the weakly-coupled system. Three-components vortex solitons (which do not feature SSB) are studied too. Because the fractional GVD breaks the system's Galilean invariance, we also address mobility of the vortex solitons, by applying a boost to them., Comment: 27 pages, 12 figures, to be published in Physica D

We study numerically the nonintegrable dynamics of coherent, solitonic, nonlinear waves, in a spatially nonlocal nonlinear Schrodinger equation relevant to realistic modelling of optical systems: the Schrodinger-Helmholtz equation. We observe a single oscillating, coherent solitary wave emerging from a variety of initial conditions. Using the direct scattering transform of the (integrable) cubic nonlinear Schrodinger equation, we find that this structure is a bound state, comprising of a primary and secondary soliton whose amplitudes oscillate out of phase. We interpret this as the solitons periodically exchanging mass. We observe this bound state self-organising from a state of incoherent turbulence, and from solitonic structures launched into the system. When a single (primary) solitonic structure is launched, a resonance process between it and waves in the system generates the secondary soliton, resulting in the bound state. Further, when two solitons are initially launched, we show that they can merge if their phases are synchronised when they collide. When the system is launched from a turbulent state comprised of many initial solitons, we propose that the bound state formation is preceded a sequence of binary collisions, in which the mass is transferred on average from the weak soliton to the strong one, with occasional soliton mergers. Both processes lead to increasingly stronger and fewer dominant solitons. The final state -- a solitary double-solitonic bound state surrounded by weakly nonlinear waves -- is robust and ubiquitous. We propose that for nonlocal media, it is a more typical statistical attractor than a single-soliton attractor suggested in previous literature., Comment: 13 figures

The well-known 1089 trick reflects an amazing trait of digital reversal process and reminisces of a limiting attractor in dynamical systems even though it takes only two steps. It is natural to consider the situations when the number of digits is beyond three as in the original 1089 trick, as well as situations when the number of steps is beyond two. The first part has been mostly done by Webster which we will reproduce. After two steps, the resulting integers are called Papadakis-Webster integers (PWI), which is always divisible by 99, and the resulting quotients consist of only 0's and 1's, which we name Papadakis-Webster binary strings (PWBS). Not all binary strings could be PWBS, and we define the hairpin pairing rule to determine if a binary string is a PWBS. For the second part, we propose a two-option iteration system named iterative digital reversal (IDR) suitably interweaving additions and subtractions. The simplest limiting behavior of IDR is 2-cycles. The elements in an IDR 2-cycle are all composed of repetitions of the 10(9)$_L$89 (L>=0) motif, and are all PWIs. The lower 2-cycle elements after division of 99 belong to the subset of PWBS that are palindromic and consist of 0- and 1-blocks with a minimal length of two. IDR also has higher p-cycles (p=10,12,71) whose elements seem to contain at least one PWI. Another interesting finding about IDR is that it contains non-periodic and diverging trajectories, as the integer values grow to infinity. In these diverging trajectories, while the number of flanking digits around the middle point increases by the iteration, the middle part has an 8-cycle rhythm or signature which has been found in all diverging trajectories. Overall, the generalization of the original 1089 trick in both space and time leads to new patterns in integers and new phenomenology in dynamics., Comment: 1 figure

We theoretically and numerically study the evolution of soliton-like waves supported by stimulated Brillouin scattering. First, the emergence and unusual behaviour of resonant solitary waves are investigated for both backward and forward three wave interactions. We find that these waves can be characterized by the ratio between the optical and acoustic damping coefficients. We also examine a second class of non-resonant anti-symmetric soliton-like waves, which have a more complicated pulse shape than traditional solitons. These waves are superluminal, with pulse velocities that can be tuned by the input Stokes and pump fields. We discuss the excitation of these types of waves and the physical conditions required for their observation., Comment: 9 pages, 7 figures

We demonstrate the possibility of stopping the soliton pulses in optical waveguide with varying and constant parameters exhibiting a Kerr nonlinear response. By using the similarity transformation, we have found the constraint condition for varying waveguide parameters and derive the exact analytical solutions for self-similar bright, kink, dark and rectangular solitary and periodic waves for nonlinear Schr\"{o}dinger equation with variable coefficients. All these generalized wave solutions depend on five arbitrary parameters and two free integration constants. It is found that the velocity of solitons is related to a free parameter $q$, which play an important role in the dynamic behavior of soliton's evolution. The precise expression of soliton's velocity shows that the solitons can be nearly stopped for appropriate values of free parameter $q$. The possibility for stopping of soliton pulses can also be realized when all parameters of nonlinear Schr\"{o}dinger equation are constant. The numerical simulations show that the stopping behavior of solitons and rectangular solitary waves can be achieved for appropriate values of free parameters characterizing these wave solutions., Comment: 10 pages, 5 figures

Far-from-equilibrium phenomena unveil the intricate principles of complex systems, including snowflake growth and fluid turbulence, with broad applications ranging from foreign exchange trading to climate modeling. A recurring feature across these systems is the emergence of a spectral cascade, where energy is transferred across the system's length scales, following a simple power law. The statistical theory of weak wave turbulence, in which only leading order interactions are considered, successfully predicts scaling laws for stationary states in idealised scenarios. Realistic conditions, such as finite size and amplitude effects, and strong dissipation, remain beyond our current understanding. Lacking comprehensive theoretical insight, we experimentally trace the formation of a spectral cascade under these conditions. Using an externally driven fluid-fluid interface, we successfully resolve individual wave modes and track their real-time evolution from one to few to many. This process culminates in a steady state whose power spectral density is fully characterised by a power-law scaling. We further quantify specific interactions through statistical correlations to reveal a hierarchy in the wave-mixing order, thus confirming a key assumption of weak-wave turbulence. We present a comprehensive time-evolution analysis that is crucial in identifying critical points where the interface undergoes significant changes. Our findings validate that the interfacial dynamics can be effectively modelled using a weakly nonlinear Lagrangian theory, enabling us to explore its applicability to other out-of-equilibrium systems. Notably, we uncover intriguing connections to reheating scenarios following cosmic inflation in the early universe.

Controlling and understanding phenomena in coupled systems remains a significant challenge across diverse fields. This study investigates a simple globally coupled chemical system that exhibits a range of rich collective dynamics, from coherence to chimera states, controllable by a system parameter. We explore the effects of altering this control parameter using two different protocols. In protocol-I, a continuous variation of the control parameter leads to memory effects, resulting in two distinct sets of phenomena in the forward and reverse directions. In contrast, protocol-II, which includes a resetting feature, yields an entirely distinct collection of behaviors. Additionally, we capture the evolution of key elements of nonequilibrium thermodynamics associated with these dynamical states under both protocols, revealing distinct signatures for each. This work highlights the complexity and uniqueness of regulating coupled systems via control parameters compared to corresponding single systems, and it carries potential implications for the manipulation and application of collective behaviors.

Waveguides subject to spatiotemporal modulations are known to exhibit nonreciprocal vibration transmission, whereby interchanging the locations of the source and receiver change the end-to-end transmission characteristics. The scenario of typical interest is unidirectional transmission in long, weakly modulated systems: when transmission is possible in one direction only. Here, with a view toward expanding their potential application as devices, we explore the vibration characteristics of spatiotemporally modulated systems that are short and strongly modulated. Focusing on two coupled systems, we develop a methodology to investigate the nonreciprocal vibration characteristics of both weakly and strongly modulated systems. In particular, we highlight the contribution of phase to nonreciprocity, a feature that is often overlooked. We show that the difference between the transmitted phases is the main contributor to breaking reciprocity in short systems. We clarify the roles of primary and side-band resonances, and their overlaps, in breaking reciprocity. We discuss the influence of modulation amplitude and wavenumber on the resonances of the modulated system.

We investigate wavelet-like localized solutions in nonlinear waveguides, enabled by complementary propagation constants embedded in domains of anomalous dispersion. They are carrier-envelope-phase stable and independent of fine details of the dispersion profile. Their spectra extend over vast ranges and they are also robust against nonlinear perturbations, providing properties highly demanded in ultrafast science. In the limits of low and high energies, approximate representations are given by modified Morlet wavelets and Gaussian pulses, respectively. We demonstrate their special features by exploiting general principles of quantum scattering theory and solid-state physics, realizing interlocked chains of wavelets analog to photonic time crystals.

In this opinion article, we gathered some reflections and practical tips on what Early Stage Researchers do against inequalities in academia. This is the longer version of an opinion paper that was recently published on the website of the International Society for Optics and Photonics (spie.org), containing more details and suggestions., Comment: comments and feedback are welcome

We discover a novel class of dissipative light bullets whose spatial profiles may be drastically reshaped along the bulk, edges and corners of the Su-Schrieffer-Heeger lattice owing to the dissipative and topological nature of the system. These light bullets appear due to resonances with different modes in lattice spectrum and may have very rich shapes that can be adiabatically controlled by varying the frequency of the external laser source. We report on robust stationary bullets and breathers which are understood from the corresponding bifurcation analysis. Our results provide a new route to realisation of reconfigurable and robust 3D light forms in topologically nontrivial systems., Comment: 7 pages, 4 figures, comments welcome

Atomistic model calculations recently revealed that the polarization pattern at the cross-section through nanoscale ferroelectric nanodomains in rhombohedral barium titanate shows a clear antiskyrmion attributes such as the net invariant topological charge of minus two. The present work confirms that these topological defects also exist as local minima in the discretized Ginzburg-Landau-Devonshire model, which is widely used in the phase-field modeling studies. We explore by phase-field simulations how an electric field can be used to create or move antiskyrmions within a monodomain ferroelectric matrix. The process of the field-induced antiskyrmion annihilation reveals the existence of a discrete ladder of intermediate antiskyrmion states with distinct radii but common symmetry, topology and geometrical properties., Comment: 5 pages, 4 figures

In an earlier work by a subset of the present authors, the method of the so-called neural deflation was introduced towards identifying a complete set of functionally independent conservation laws of a nonlinear dynamical system. Here, we extend by a significant step this proposal. Instead of using the explicit knowledge of the underlying equations of motion, we develop the method directly from system trajectories. This is crucial towards enhancing the practical implementation of the method in scenarios where solely data reflecting discrete snapshots of the system are available. We showcase the results of the method and the number of associated conservation laws obtained in a diverse range of examples including 1D and 2D harmonic oscillators, the Toda lattice, the Fermi-Pasta-Ulam-Tsingou lattice and the Calogero-Moser system.

Dryland vegetation can survive water stress by forming spatial patterns but is often subjected to herbivory as an additional stress that puts it at risk of desertification. Understanding the mutual relationships between vegetation patterning and herbivory is crucial for securing food production in drylands, which constitute the majority of rangelands worldwide. Here, we introduce a novel vegetation-herbivore model that captures pattern-forming feedbacks associated with water and herbivory stress and a behavioral aspect of herbivores representing an exploitation strategy.Applying numerical continuation methods, we analyze the bifurcation structure of uniform and patterned vegetation-herbivore solutions, and use direct numerical simulations to study various forms of collective herbivore dynamics. We find that herbivory stress can induce traveling vegetation-herbivore waves and uncover the ecological mechanism that drives their formation. In the traveling-wave state, the herbivore distribution is asymmetric with higher density on one side of each vegetation patch. At low precipitation values their distribution is localized, while at high precipitation the herbivores are spread over the entire landscape. Importantly, their asymmetric distribution results in uneven herbivory stress, strong on one side of each vegetation patch and weak on the opposing side - weaker than the stress exerted in spatially uniform herbivore distribution. Consequently, the formation of traveling waves results in increased sustainability to herbivory stress. We conclude that vegetation-herbivore traveling waves may play an essential role in sustaining herbivore populations under conditions of combined water and herbivory stress, thereby contributing to food security in endangered regions threatened by droughts and population growth., Comment: 16 pages, 13 figures

We explore the chaotic dynamics of a large one-dimensional lattice of coupled maps with diffusive coupling of varying strength using the covariant Lyapunov vectors (CLVs). Using a lattice of diffusively coupled quadratic maps we quantify the growth of spatial structures in the chaotic dynamics as the strength of diffusion is increased. When the diffusion strength is increased from zero, we find that the leading Lyapunov exponent decreases rapidly from a positive value to zero to yield a small window of periodic dynamics which is then followed by chaotic dynamics. For values of the diffusion strength beyond the window of periodic dynamics, the leading Lyapunov exponent does not vary significantly with the strength of diffusion with the exception of a small variation for the largest diffusion strengths we explore. The Lyapunov spectrum and fractal dimension are described analytically as a function of the diffusion strength using the eigenvalues of the coupling operator. The spatial features of the CLVs are quantified and compared with the eigenvectors of the coupling operator. The chaotic dynamics are composed entirely of physical modes for all of the conditions we explore. The leading CLV is highly localized and the localization decreases with increasing strength of the spatial coupling. The violation of the dominance of Oseledets splitting indicates that the entanglement of pairs of CLVs becomes more significant between neighboring CLVs as the strength of the diffusion is increased., Comment: 33 pages, 15 figures

We explore the velocity distributions in a vibrated binary granular gas system, focusing on how these distributions are influenced by the coefficient of restitution (CoR) and the inelasticity of particle collisions. Through molecular dynamics simulations, we examine the system's behavior for a range of CoR values below unity: specifically, 0.80, 0.85, 0.90, and 0.95. We track the evolution of velocity distributions as the system approaches a non-equilibrium steady state. Our findings reveal a clear departure from the classical Maxwell-Boltzmann distribution, with increasing deviations as CoR decreases, indicating non-equipartition of energy between the two particle types. This behavior underscores the intricate dynamics inherent in inelastic collisions and highlights the significance of particle inelasticity in determining the system's velocity distributions. These results provide insight into non-equilibrium statistical mechanics and have implications for real-world applications where granular materials are subject to external vibrations.

We study the 1d swarmalator model in the continuum limit. We examine the stability of its collective states which have compact support: synchrony, where the swarmalators lie in two sync dots (zero dimensional support), and the phase wave, where the swarmalators line up in a ring with uniformly spaced positions and phases (one dimensional support). The compact support imposes analytic difficulties that occur in many other swarmalator models and thus is blocking progress in the field. We show how to overcome this difficulty, deriving the two states' stability spectra exactly.

We study the bifurcation of planar patterns and fast-moving pattern interfaces in an asymptotic long-wave model for the three-dimensional B\'enard-Marangoni problem, which is close to a Turing instability. We derive the model from the full free-boundary B\'enard-Marangoni problem for a thin liquid film on a heated substrate of low thermal conductivity via a lubrication approximation. This yields a quasilinear, fully coupled, mixed-order degenerate-parabolic system for the film height and temperature. As the Marangoni number $M$ increases beyond a critical value $M^*$, the pure conduction state destabilises via a Turing(-Hopf) instability. Close to this critical value, we formally derive a system of amplitude equations which govern the slow modulation dynamics of square or hexagonal patterns. Using center manifold theory, we then study the bifurcation of square and hexagonal planar patterns. Finally, we construct planar fast-moving modulating travelling front solutions that model the transition between two planar patterns. The proof uses a spatial dynamics formulation and a center manifold reduction to a finite-dimensional invariant manifold, where modulating fronts appear as heteroclinic orbits. These modulating fronts facilitate a possible mechanism for pattern formation, as previously observed in experiments., Comment: 55 pages, 18 figures, Supplementary Material available at https://github.com/Bastian-Hilder/TuringUnstableThinFilmFronts

We apply spatial dynamical-systems techniques to prove that certain spatiotemporal patterns in reversible reaction-diffusion equations undergo snaking bifurcations. That is, in a narrow region of parameter space, countably many branches of patterned states coexist that connect at towers of saddle-node bifurcations. Our patterns of interest are contact defects, which are 1-dimensional time-periodic patterns with a spatially oscillating core region that at large distances from the origin in space resemble pure temporally oscillatory states and arise as natural analogues of spiral and target waves in one spatial dimension. We show that these solutions lie on snaking branches that have a more complex structure than has been seen in other contexts. In particular, we predict the existence of families of asymmetric travelling defect solutions with arbitrary background phase offsets, in addition to symmetric standing target and spiral patterns. We prove the presence of these additional patterns by reconciling results in classic ODE studies with results from the spatial-dynamics study of patterns in PDEs and using geometrical information contained in the stable and unstable manifolds of the background wavetrains and their natural equivariance structure., Comment: 32 pages, 11 figures

We numerically study the nonlinear interactions of high-frequency circularly polarized electromagnetic (EM) waves and low-frequency electron-acoustic (EA) density perturbations driven by the EM wave ponderomotive force in relativistic plasmas with two groups of electrons--the population of relativistic degenerate dense electrons (bulk plasma) and the sparse relativistic nondegenerate (classical) electrons, and immobile singly charged positive ions. By pattern selection, we show that many solitary patterns can be generated and drenched through modulational instability of EM waves at different spatial length scales and that the EM wave radiation spectra emanating from compact astrophysical objects may not settle into stable envelope solitons but into different incoherent states, including the emergence of temporal and spatiotemporal chaos due to collisions and fusions among the patterns with strong EA wave emission. The appearance of these states is confirmed by analyzing the Lyapunov exponent spectra, correlation function, and mutual information. As a result, the redistribution of wave energy from initially excited many solitary patterns at large scales to a few new incoherent patterns with small wavelengths in the system occurs, leading to the onset of turbulence in astrophysical plasmas., Comment: 13 pages, 13 figures

Bistable mechanical metamaterials have shown promise for mitigating the harmful consequences of impact by converting kinetic energy into stored strain energy, offering an alternative and potentially synergistic approach to conventional methods of attenuating energy transmission. In this work, we numerically study the dynamic response of a one-dimensional bistable metamaterial struck by a high speed impactor (where the impactor velocity is commensurate with the sound speed), using the peak kinetic energy experienced at midpoint of the metamaterial compared to that in an otherwise identical linear system as our performance metric. We make five key findings: 1) The bistable material can counter-intuitively perform better (to nearly 1000x better than the linear system) as the viscosity decreases (but remains finite), but only when sufficiently fine discretization has been reached (i.e. the system approaches sufficiently close to the continuum limit); 2) This discretization threshold is sharp, and depends on the viscosity present; 3) The bistable materials can also perform significantly worse than linear systems (for low discretization and viscosity or zero viscosity); 4) The dependence on discretization stems from the partition of energy into trains of solitary waves that have pulse lengths proportional to the unit cell size, where, with intersite viscosity, the solitary wave trains induce high velocity gradients and thus enhanced damping compared to linear, and low-unit-cell-number bistable, materials; and 5) When sufficiently fine discretization has been reached at low viscosities, the bistable system outperforms the linear one for a wide range of impactor conditions. The first point is particularly important, as it shows the existence of a nonlinear dynamical size effect, where, given a protective layer of some thickness...

We study equivariant wave maps from the $2+1$ dimensional wormhole to the 2-sphere. This model has explicit harmonic map solutions which, in suitable coordinates, have the form of the sine-Gordon kinks/anti-kinks. We conjecture that there exist asymptotically static chains of $N\geq 2$ alternating kinks and anti-kinks whose subsequent rates of expansion increase in geometric progression as $t\rightarrow \infty$. Our argument employs the method of collective coordinates to derive effective finite-dimensional ODE models for the asymptotic dynamics of $N$-chains. For $N=2,3$ the predictions of these effective models are verified by direct PDE computations which demonstrate that the $N$-chains lie at the threshold of kink-anti-kink annihilation., Comment: 16 pages, 6 figures

Auto-chemotaxis, the directed movement of cells along gradients in chemicals they secrete, is central to the formation of complex spatiotemporal patterns in biological systems. Since the introduction of the Keller-Segel model, numerous variants have been analyzed, revealing phenomena such as coarsening of aggregates, stable aggregate sizes, and spatiotemporally chaotic dynamics. Here, we consider general mass-conserving Keller-Segel models, that is, models without cell growth and death, and analyze the generic long-time dynamics of the chemotactic aggregates. Building on and extending our previous work, which demonstrated that chemotactic aggregation can be understood through a generalized Maxwell construction balancing density fluxes and reactive turnover, we use singular perturbation theory to derive the full rates of mass competition between well-separated aggregates. We analyze how this mass-competition process drives coarsening in both diffusion- and reaction-limited regimes, with the diffusion-limited rate aligning with our previous quasi-steady-state analyses. Our results generalize earlier mathematical findings, demonstrating that coarsening is driven by self-amplifying mass transport and aggregate coalescence. Additionally, we provide a linear stability analysis of the lateral instability, predicting it through a nullcline-slope criterion similar to the curvature criterion in spinodal decomposition. Overall, our findings suggest that chemotactic aggregates behave similarly to phase-separating droplets, providing a robust framework for understanding the coarse-grained dynamics of auto-chemotactic cell populations and providing a basis for the analysis of more complex multi-species chemotactic systems., Comment: 24 pages, 5 figures

Chemotaxis allows single cells to self-organize at the population level, as classically described by Keller-Segel models. We show that chemotactic aggregation can be understood using a generalized Maxwell construction based on the balance of density fluxes and reactive turnover. This formulation implies that aggregates generically undergo coarsening, which is interrupted and reversed by cell growth and death. Together, both stable and spatiotemporally dynamic aggregates emerge. Our theory mechanistically links chemotactic self-organization to phase separation and reaction-diffusion patterns., Comment: 7 pages Main text, 25 pages Supplementary Information; 4 figures

For cellular functions like division and polarization, protein pattern formation driven by NTPase cycles is a central spatial control strategy. Operating far from equilibrium, no general theory links microscopic reaction networks and parameters to the pattern type and dynamics. We discover a generic mechanism giving rise to an effective interfacial tension organizing the macroscopic structure of non-equilibrium steady-state patterns. Namely, maintaining protein-density interfaces by cyclic protein attachment and detachment produces curvature-dependent protein redistribution which straightens the interface. We develop a non-equilibrium Neumann angle law and Plateau vertex conditions for interface junctions and mesh patterns, thus introducing the concepts of ``Turing mixtures'' and ``Turing foams''. In contrast to liquid foams and mixtures, these non-equilibrium patterns can select an intrinsic wavelength by interrupting an equilibrium-like coarsening process. Data from in vitro experiments with the E. coli Min protein system verifies the vertex conditions and supports the wavelength dynamics. Our study uncovers interface laws with correspondence to thermodynamic relations that arise from distinct physical processes in active systems. It allows the design of specific pattern morphologies with potential applications as spatial control strategies in synthetic cells., Comment: 11 pages main text, 5 pages Methods, 64 pages Supplementary Information; 12 figures

We study the propagation of a domain wall (kink) of the $\phi^4$ model in a radially symmetric environment defined by a gravity source. This source deforms the standard Euclidian metric into a Schwartzschild-like one. We introduce an effective model that accurately describes the dynamics of the kink center. This description works well even outside the perturbation region, i.e., even for large masses of the gravitating object. We observed that such a spherical domain wall surrounding a star-type object inevitably "collapses", i.e., shrinks in radius towards the origin and offer an understanding of the latter phenomenology. The relevant analysis is presented for a circular domain wall and a spherical one., Comment: 19 pages, 12 figures

In the rapidly rotating limit, we derive a balanced set of reduced equations governing the strongly nonlinear development of the convective wall-mode instability in the interior of a general container. The model illustrates that wall-mode convection is a multiscale phenomenon where the dynamics of the bulk interior diagnostically determine the small-scale dynamics within Stewartson boundary layers at the sidewalls. The sidewall boundary layers feedback on the interior via a nonlinear lateral heat-flux boundary condition, providing a closed system. Outside the asymptotically thin boundary layer, the convective modes connect to a dynamical interior that maintains scales set by the domain geometry. In many ways, the final system of equations resembles boundary-forced planetary geostrophic baroclinic dynamics coupled with barotropic quasi-geostrophic vorticity. The reduced system contains the results from previous linear instability theory but captured in an elementary fashion, providing a new avenue for investigating wall-mode convection in the strongly nonlinear regime. We also derive the dominant Ekman-flux correction to the onset Rayleigh number for large Taylor number, Ra ~ 31.8 Ta^(1/2) - 4.43 Ta^(5/12) for no-slip boundaries. We demonstrate some of the reduced model's nonlinear dynamics with numerical simulations in a cylindrical container., Comment: 38 pages, 13 figures

We investigate the dynamics arising out of the propagation of light pulses with different polarizations through a condensate (referred to as a constant background field) with cross coupling described by a coupled nonlinear Schrodinger equation(NLSE) type equation. We then employ Gauge and Darboux transformation approach to bring out the rich dynamics arising out of the background field and cross coupling. The collisional dynamics of bright solitons is found to be inelastic. The constant background field is found to facilitate the periodic localization of light pulses during propagation. We have also unearthed breathers, bright-bright, bright-dark and dark-bright solitons of the coupled NLSE. While the amplitude of breathers oscillate with time as predicted, their maximum(or minimum) amplitude is found to remain a constant and the addition of cross coupling only contributes to the rapid fluctuations in its amplitude over a period of time. In addition, the reinforcement of cross coupling in the presence of constant wave field facilitates the interference of light pulses leading to interesting pattern formation among bright-bright, bright-dark and dark-bright solitons. The highlight of the results is that one obtains various localized excitations like breathers, bright and dark solitons by simply manipulating the amplitude of the constant wave field., Comment: 14 pages, 6 figures, Accepted for Publication in Romanian Reports in Physics (2024)

It has recently been theoretically predicted and experimentally observed that a soliton resulting from nonlinearity can be pumped across an integer or fractional number of unit cells as a system parameter is slowly varied over a pump period. Nonlinear Thouless pumping is now understood as the flow of instantaneous Wannier functions, ruling out the possibility of pumping a soliton across a nonzero number of unit cells over one cycle when a corresponding Wannier function does not exhibit any flow, i.e., when the corresponding Bloch band that the soliton bifurcates from is topologically trivial. Here we surprisingly find an anomalous nonlinear Thouless pump where the displacement of a soliton over one cycle is twice the Chern number of the Bloch band from which the soliton bifurcates. We show that the breakdown of the correspondence between the displacement of a soliton and the Chern number arises from the emergence of a new branch of stable soliton solutions mainly consisting of two neighboring instantaneous Wannier functions. Furthermore, we find a nonlinearity-induced integer quantized Thouless pump, allowing a soliton to travel across one unit cell during a pump period, even when the corresponding band is topologically trivial. Our results open the door to studying nonlinearity-induced Thouless pumping of solitons., Comment: 7+8 pages, 4+4 figures

Contemporary epidemiological models often involve spatial variation, providing an avenue to investigate the averaged dynamics of individual movements. In this work, we extend a recent model by Vaziry, Kolokolnikov, and Kevrekidis [Royal Society Open Science 9 (10), 2022] that included, in both infected and susceptible population dynamics equations, a cross-diffusion term with the second spatial derivative of the infected population density. Diffusion terms of this type occur, for example, in the Keller-Siegel chemotaxis model. The presented model corresponds to local orderly commute of susceptible and infected individuals, and is shown to arise in two dimensions as a limit of a discrete process. The present contribution identifies and studies specific features of the new model's dynamics, including various types of infection waves and buffer zones protected from the infection. The model with vital dynamics additionally exhibits complex spatiotemporal behaviour that involves the generation of quasiperiodic infection waves and emergence of transient strongly heterogeneous patterns.

In this work, we study the evolution of disturbances within the framework of the Cubic Vortical Whitham (CV-Whitham) equation, considering both positive and negative cubic nonlinearities. This equation plays important role for description of the wave processes in the presence of shear flows. We find well-formed breather-type structures arising from the evolution of depression disturbances with positive cubic nonlinearity. For elevation disturbances, the results are two-fold. When the cubic nonlinearity is negative, we show that the CV-Whitham equation and the Gardner equation are qualitatively similar, differing only by a small phase lag due to differences in the dispersion term. However, with positive cubic nonlinearity, the differences between the solutions become more pronounced, with the CV-Whitham equation producing sharper waves that suggest the onset of wave breaking.

We study the properties of modulational instability and discrete breathers arising in a quasi-one-dimensional discrete Gross-Pitaevskii equation with Lee-Huang-Yang corrections. Conditions for modulation instability and instability regions of nonlinear plane waves are determined in parameter space. We analytically investigate the existence of different quantum droplet solutions, including intersite, onsite, front-like, flat-top and dark localized modes, using the Page method and variational approach. Their stability is checked using linear stability analyses and numerical simulations. The analytical predictions corroborated with the numerical simulations., Comment: 12 pages, 21 figures

We report the observation of a two-dimensional dam break flow of a photon fluid in a nonlinear optical crystal. By precisely shaping the amplitude and phase of the input wave, we investigate the transition from one-dimensional (1D) to two-dimensional (2D) nonlinear dynamics. We observe wave breaking in both transverse spatial dimensions with characteristic timescales determined by the aspect ratio of the input box-shaped field. The interaction of dispersive shock waves propagating in orthogonal directions gives rise to a 2D ensemble of solitons. Depending on the box size, we report the evidence of a dynamic phase characterized by a constant number of solitons, resembling a 1D solitons gas in integrable systems. We measure the statistical features of this gaseous-like phase. Our findings pave the way to the investigation of collective solitonic phenomena in two dimensions, demonstrating that the loss of integrability does not disrupt the dominant phenomenology.

We study the dynamics of solitons under the action of one-dimensional quasiperiodic lattice potentials, fractional diffraction, and nonlinearity. The formation and stability of the solitons is investigated in the framework of the fractional nonlinear Schr\"odinger equation. By means of variational and numerical methods, we identify conditions under which stable solitons emerge, stressing the effect of the fractional diffraction on soliton properties. The reported findings contribute to the understanding of the soliton behavior in complex media, with implications for topological photonics and matter-wave dynamics in lattice potentials., Comment: 14 pages, 19 figures