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The aim of this paper is to explore a Schrödinger equation that incorporates a higher‐order operator. Traditional models for electron dynamics have utilized a second‐order diffusion Schrödinger equation, where oscillatory behavior is achieved through complex domain formulations. Incorporating a higher‐order operator enables the induction of oscillatory spatial patterns in solutions. Our analysis initiates with a variational formulation within generalized spaces, facilitating the examination of solution boundedness. Subsequently, we delve into the oscillatory characteristics of solutions, drawing upon a series of lemmas originally applied to the Kuramoto–Sivashinsky equation, the Cahn–Hilliard equation, and other equations that employ higher‐order operators. Specific solution types, such as standing waves, are numerically investigated to illustrate the oscillatory spatial patterns. The discussion then extends to the theory of traveling waves to establish general conditions for positive solutions. A contribution of this work is the precise evaluation of a critical traveling wave speed, denoted as c∗$$ {c}^{\ast } $$, above which the first minimum remains positive. For values of the traveling wave speed c$$ c $$ significantly greater than c∗$$ {c}^{\ast } $$, the solutions can be entirely positive. [ABSTRACT FROM AUTHOR]

We study the dynamics of waves, oscillations, and other spatio-temporal patterns in stochastic evolution systems, including SPDE and stochastic integral equations. Representing a given pattern as a smooth, stable invariant manifold of the deterministic dynamics, we reduce the stochastic dynamics to a finite dimensional SDE on this manifold using the isochronal phase. The isochronal phase is defined by mapping a neighborhood of the manifold onto the manifold itself, analogous to the isochronal phase defined for finite-dimensional oscillators by A.T. Winfree and J. Guckenheimer. We then determine a probability measure that indicates the average position of the stochastic perturbation of the pattern/wave as it wanders over the manifold. It is proved that this probability measure is accurate on time-scales greater than O (σ − 2) , but less than O (exp ⁡ (C σ − 2)) , where σ ≪ 1 is the amplitude of the stochastic perturbation. Moreover, using this measure, we determine the expected velocity of the difference between the deterministic and stochastic motion on the manifold. [ABSTRACT FROM AUTHOR]

This research employs the improved modified extended tanh-function technique to explore several solitary wave solutions to the fractional-order Fokas equation. The propagation of waves in fluid dynamics and optical systems are two examples of various natural phenomena that are effectively addressed by the fractional-order Fokas equation. The model captures a generalization of the integer derivative form by including fractional derivatives defined in the conformable sense. We use the phase portrait theory to investigate the existence of traveling wave solutions. The improved modified extended tanh-function technique is successfully applied as a reliable analytical procedure to derive several solitary wave solutions, providing an approachable structure to deal with the complexity introduced by the fractional order. The extracted solutions, which are illustrated by hyperbolic, trigonometric, and rational functions, exhibit a variety of solitary wave shapes, such as bell-shaped, kink, and anti-kink patterns. We additionally evaluate how well the employed method performs in comparison to other approaches. Furthermore, some graphical visualizations are provided to clearly demonstrate the physical behavior of the obtained solutions under various parameter values. The outcomes highlight the effectiveness and adaptability of the proposed strategy in resolving fractional nonlinear differential equations and expand our knowledge of fractional-order systems. [ABSTRACT FROM AUTHOR]

In this paper, we study traveling waves for a time-delayed nonlocal reaction-diffusion model of within-host viral infections. Firstly, we establish the existence of semi-traveling waves that converge to an unstable infection-free equilibrium as the moving coordinate goes to −∞, provided the wave speed c > c ⁎ for some positive number c ⁎ and the basic reproduction number R 0 > 1. Then we construct a Lyapunov functional to show that the semi-traveling waves converge to an endemic equilibrium as the moving coordinate goes to +∞, and use a limiting argument to obtain the existence of the traveling wave connecting these two equilibria for c = c ⁎ and R 0 > 1. We further employ a Laplace transform technique to prove the non-existence of bounded semi-traveling waves when 0 < c < c ⁎ and R 0 > 1. It turns out that c ⁎ is the minimum wave speed for traveling waves connecting the infection-free equilibrium and the endemic equilibrium. Finally, we conduct numerical simulations to illustrate the long-time behavior of solutions and the dependence of c ⁎ on parameters. [ABSTRACT FROM AUTHOR]

The FitzHugh–Nagumo equation, originally conceived in neuroscience during the 1960s, became a key model providing a simplified view of excitable neuron cell behavior. Its applicability, however, extends beyond neuroscience into fields like cardiac physiology, cell division, population dynamics, electronics, and other natural phenomena. In this review spanning six decades of research, we discuss the diverse spatio-temporal dynamical behaviors described by the FitzHugh–Nagumo equation. These include dynamics like bistability, oscillations, and excitability, but it also addresses more complex phenomena such as traveling waves and extended patterns in coupled systems. The review serves as a guide for modelers aiming to utilize the strengths of the FitzHugh–Nagumo model to capture generic dynamical behavior. It not only catalogs known dynamical states and bifurcations, but also extends previous studies by providing stability and bifurcation analyses for coupled spatial systems. [ABSTRACT FROM AUTHOR]

While previous works established the inhibitory role of alpha oscillations during working memory maintenance, it remains an open question whether such an inhibitory control is a top-down process. Here, we attempted to disentangle this issue by considering the spatiotemporal component of waves in the alpha band, i.e., alpha traveling waves. We reanalyzed two pre-existing and open-access EEG datasets (N = 180, 90 males, 80 females, 10 unknown) where participants performed lateralized, visual delayed match-to-sample working memory tasks. In the first dataset, the distractor load was manipulated (2, 4, or 6), whereas in the second dataset, the memory span varied between 1, 3, and 6 items. We focused on the propagation of alpha waves on the anterior-posterior axis during the retention period. Our results reveal an increase in alpha-band forward waves as the distractor load increased, but also an increase in forward waves and a decrease in backward waves as the memory set size increased. Our results also showed a lateralization effect: alpha forward waves exhibited a more pronounced increase in the hemisphere contralateral to the distractors, whereas the reduction in backward waves was stronger in the hemisphere contralateral to the targets. In short, the forward waves were regulated by distractors, whereas targets inversely modulated backward waves. Such a dissociation of goal-related and goal-irrelevant physiological signals suggests the coexistence of bottom-up and top-down inhibitory processes: alpha forward waves might convey a gating effect driven by distractor load, while backward waves may represent direct top-down gain control of downstream visual areas. [ABSTRACT FROM AUTHOR]

In this paper we study traveling wave solutions to the free boundary incompressible Navier-Stokes system with generalized Navier-slip conditions. The fluid is assumed to occupy a horizontally infinite strip-like domain that is bounded below by a flat rigid surface and above by a moving surface. We assume that the fluid is acted upon by a bulk force and a surface stress that are stationary in a coordinate system moving parallel to the fluid bottom, and a uniform gravitational force that is perpendicular to the flat rigid surface. We construct our solutions via an implicit function argument, and show that as the slip parameter shrinks to zero, the Navier-slip solutions converge to solutions to the no-slip problem obtained previously. [ABSTRACT FROM AUTHOR]

This paper is concerned with the traveling waves for a modified Holling-Tanner predator–prey model with degenerate diffusion. Different from the established approaches of constructing upper-lower solutions, we construct a new and suitable pair of upper-lower solutions by solving three differential equations and establish the existence of traveling waves for any c ≥ c ∗ when n ≥ 0 . In addition, we obtain the minimal wave speed c ∗ = 2 r when n = 0 , without reconstructing the upper-lower solutions. Furthermore, the asymptotic behavior of traveling waves at infinity is obtained by the upper-lower solutions and the contracting rectangle method. [ABSTRACT FROM AUTHOR]

The general solutions for nonlinear waves in a bubble-liquid mixture are obtained from the Kudryashov–Sinelshchikov (KS) equation under different parametric regimes. The Chiellini integrability condition has been used for constructing exact general solutions. In the non-dissipative case, we find that the solutions may be expressed in terms of Jacobi elliptic and Weierstrass functions. On the other hand, in the dissipative case, only implicit solutions are obtained for the KS equation. For an alternative perspective, the notion of the Jacobi Last Multiplier has been utilized to obtain the corresponding Lagrangian and Hamiltonian description of the reduced equation for the bubble-liquid mixture system. [ABSTRACT FROM AUTHOR]

We obtain the exact analytical traveling wave solutions of the Kolmogorov–Petrovskii–Piskunov equation, with the reaction term belonging to the class of functions, which includes that of the (generalized) Fisher equation, for the particular values of the wave's speed. Additionally we obtain the exact analytical traveling wave solutions of the generalized Burgers–Huxley equation. [ABSTRACT FROM AUTHOR]

Foam flow in porous media increased the scientific community's attention due to several potential industrial applications, including remediation of contaminated aquifers, soil remediation, acid diversion, and hydrocarbon recovery. Natural reservoirs typically have fractured and multi-layered structures. We investigate an immiscible incompressible two-phase foam flow in an internally homogeneous two-layered porous medium with different porosities and absolute permeabilities. For our analysis, we extended the previous result, evidencing that the presence of foam induces the existence of a single flow front in both layers. Using the traveling wave solution, we classify the foam flow depending on the absolute permeability and the porosity ratio between layers. We show that the mass crossflow between layers is connected to the relative position of the flow front in these layers and that the porosity difference between layers impacts the mass crossflow. All analytical estimates were supported by direct numerical simulations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation phenomena for the nonlocal reaction-diffusion monostable equation with delay of the form ∂ u (t , x) ∂ t = d Δ u (t , x) + f (u (t , x) , ∫ R J (x − y) u (t − τ , y) d y) , t > 0 , x ∈ R. It is well-known that if we take J (x) = δ (x) and τ = 0 , there exists a minimal wave speed c ⁎ > 0 , such that this equation has no traveling wave front for 0 < c < c ⁎ and a traveling wave front for each c ≥ c ⁎ , which is unique up to translation and is globally asymptotically stable. Furthermore, when J is symmetry and exponentially bounded and τ > 0 , Wang et al. (2008) [27] considered the effects of delay and nonlocality on the spreading speed and proved that (i) if ∂ 2 f (0 , 0) > 0 , then the delay can slow the spreading speed of the wave fronts and the nonlocality can increase the spreading speed; and (ii) if ∂ 2 f (0 , 0) = 0 , then the delay and nonlocality do not affect the spreading speed. However, when J is asymmetry or exponentially unbounded , the question was left open. In this paper we obtain a rather complete answer to this question. More precisely, we show that for exponentially bounded kernels the minimal speed of traveling waves exists and coincides with the spreading speed. We also investigate the case of exponentially unbounded kernels where we prove the non-existence of traveling wave solutions and obtain upper and lower bounds for the position of any level set of the solutions. These bounds allow us to estimate how the solutions spread, depending on the kernels. [ABSTRACT FROM AUTHOR]

In this paper, we develop a hybrid approach to solve the viscous Burgers' equation by combining classical boundary layer theory with modern Physics-Informed Neural Networks (PINNs). The boundary layer theory provides an approximate analytical solution to the equation, particularly in regimes where viscosity dominates. PINNs, on the other hand, offer a data-driven framework that can address complex boundary and initial conditions more flexibly. We demonstrate that PINNs capture the key dynamics of the Burgers' equation, such as shock wave formation and the smoothing effects of viscosity, and show how the combination of these methods provides a powerful tool for solving nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we mainly devote to investigate the classification of traveling waves to the generalized Camassa–Holm equation with dual-power nonlinearities. Utilizing the celebrated approach of classification of traveling wave solution which was proposed by Jonatan Lenells [Traveling wave solutions of the Camassa-Holm equation. J Diff Equ. 2005;217(2):393–430]. We show a result concerns the regularity of traveling waves and then classify all traveling wave solutions. [ABSTRACT FROM AUTHOR]

We prove the existence of large amplitude bi-periodic traveling waves (stationary in a moving frame) of the two-dimensional non-resistive Magnetohydrodynamics (MHD) system with a traveling wave external force with large velocity speed λ (ω 1 , ω 2) and of amplitude of order O (λ 1 + ) where λ ≫ 1 is a large parameter. For most values of ω = (ω 1 , ω 2) and for λ ≫ 1 large enough, we construct bi-periodic traveling wave solutions of arbitrarily large amplitude as λ → + ∞. More precisely, we show that the velocity field is of order O (λ 0 + ) , whereas the magnetic field is close to a constant vector as λ → + ∞. Due to the presence of small divisors, the proof is based on a nonlinear Nash–Moser scheme adapted to construct nonlinear waves of large amplitude. The main difficulty is that the linearized equation at any approximate solution is an unbounded perturbation of large size of a diagonal operator and hence the problem is not perturbative. The invertibility of the linearized operator is then performed by using tools from micro-local analysis and normal forms together with a sharp analysis of high- and low-frequency regimes with respect to the large parameter λ ≫ 1. To the best of our knowledge, this is the first result in which global in time, large amplitude solutions are constructed for the 2D non-resistive MHD system with periodic boundary conditions and also the first existence result of large amplitude quasi-periodic solutions for a nonlinear PDE in higher space dimension. [ABSTRACT FROM AUTHOR]

The development of biologically realistic models of brain microcircuits and regions constitutes currently a very relevant topic in computational neuroscience. One of the main challenges of such models is the passage between different scales, going from the microscale (cellular) to the meso (microcircuit) and macroscale (region or whole-brain level), while keeping at the same time a constraint on the demand of computational resources. In this paper we introduce a multiscale modeling framework for the hippocampal CA1, a region of the brain that plays a key role in functions such as learning, memory consolidation and navigation. Ourmodeling framework goes fromthe single cell level to the macroscale and makes use of a novel mean-field model of CA1, introduced in this paper, to bridge the gap between the micro and macro scales. We test and validate the model by analyzing the response of the system to the main brain rhythms observed in the hippocampus and comparing our results with the ones of the corresponding spiking network model of CA1. Then, we analyze the implementation of synaptic plasticity within our framework, a key aspect to study the role of hippocampus in learning and memory consolidation, and we demonstrate the capability of our framework to incorporate the variations at synaptic level. Finally, we present an example of the implementation of our model to study a stimulus propagation at the macro-scale level, and we show that the results of our framework can capture the dynamics obtained in the corresponding spiking network model of the whole CA1 area. [ABSTRACT FROM AUTHOR]

Vibration testing of an integrally bladed rotor (IBR) is often completed through traveling wave excitation (TWE) bench tests composed of multiple, simultaneously excited inputs. Often, each blade has many output locations. For IBRs with many blades, as often found in the high pressure compressor, the total number of outputs can be several orders of magnitude. Formulation of a full output spectral density matrix from all measurements will then contain an exponential number of values at each frequency bin that can be detrimental to computational resources during the spectral density matrix formulation as well as down-stream system identification algorithms. An online algorithm is proposed for collecting and analyzing TWE data to reduce the large, computationally burdensome datasets into a manageable number of subsets for subsequent system identification. Furthermore, a frequency domain decomposition technique is also proposed for system identification that also attempts to reduce the data size through singular value decomposition. Identified system poles can be averaged from each subset, but mode shapes require stitching each subset together to identify the full mode shape at all output locations. The developed approaches are tested on synthetic TWE data and compared to baseline system identification results obtained using the full spectral density matrix. Results indicate the data subsets accurately compare to the baseline without much loss in accuracy. [ABSTRACT FROM AUTHOR]

This paper considers the wave propagation for a spatial discrete virus model with HIV viral load and 2-LTR dynamics during high active antiretroviral therapy. Applying Schauder fixed point theorem, technique of Lyapunov function and limiting arguments, we establish the existence of traveling wave solutions for the virus model when the basic reproduction number is larger than one and the wave speed is not less than a threshold speed. Then, using the methods of comparison principle and Laplace transform, we prove the nonexistence of traveling wave solutions when the basic reproduction number is either smaller than one; or larger than one but the wave speeds are less than the threshold speed. Indeed, the threshold speed is minimum wave speed for the propagation of traveling waves. We further show that the three classes of inhibitor can decrease the minimum wave speed; while the diffusion rate of visions will increase the minimum wave speed. [ABSTRACT FROM AUTHOR]

We consider a traveling wave solution of a two-dimensional transmission problem across the $ y- $axis modeling autoignition of reactive jets, with a limiting version corresponding to large heat loss. A jump of the normal derivative generates a singularity at the origin. We construct an explicit solution to a problem in a distributional sense that verifies the transmission system and closely examine the properties of the solution near the singularity. Using the Implicit Function Theorem, we study the level sets, especially the one through the origin. The numerical illustrations are consistent with our theoretical results. [ABSTRACT FROM AUTHOR]

In the present study, we investigated traveling waves induced by transcranial alternating current stimulation in the alpha frequency band of healthy subjects. Electroencephalographic data were recorded in 12 healthy subjects before, during, and after phase-shifted stimulation with a device combining both electroencephalographic and stimulation capacities. In addition, we analyzed the results of numerical simulations and compared them to the results of identical analysis on real EEG data. The results of numerical simulations indicate that imposed transcranial alternating current stimulation induces a rotating electric field. The direction of waves induced by stimulation was observed more often during at least 30 s after the end of stimulation, demonstrating the presence of aftereffects of the stimulation. Results suggest that the proposed approach could be used to modulate the interaction between distant areas of the cortex. Non-invasive transcranial alternating current stimulation can be used to facilitate the propagation of circulating waves at a particular frequency and in a controlled direction. The results presented open new opportunities for developing innovative and personalized transcranial alternating current stimulation protocols to treat various neurological disorders. [ABSTRACT FROM AUTHOR]

The computational resources of a neuromorphic network model introduced earlier were investigated in the first paper of this series. It was argued that a form of ubiquitous spontaneous local convolution enabled logical gate-like neural motifs to form into hierarchical feed-forward structures of the Hubel-Wiesel type. Here we investigate concomitant data-like structures and their dynamic rôle in memory formation, retrieval, and replay. The mechanisms give rise to the need for general inhibitory sculpting, and the simulation of the replay of episodic memories, well known in humans and recently observed in rats. Other consequences include explanations of such findings as the directional flows of neural waves in memory formation and retrieval, visual anomalies and memory deficits in schizophrenia, and the operation of GABA agonist drugs in suppressing episodic memories. We put forward the hypothesis that all neural logical operations and feature extractions are of the convolutional hierarchical type described here and in the earlier paper, and exemplified by the Hubel-Wiesel model of the visual cortex, but that in more general cases the precise geometric layering might be obscured and so far undetected. [ABSTRACT FROM AUTHOR]

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems. [ABSTRACT FROM AUTHOR]

This study explores the existence of traveling wave solutions in the classical Kermack-McKendrick epidemic model with local diffusive. The findings highlight the critical role of the basic reproduction number $R_0$in shaping wave dynamics. Traveling wave solutions are shown to exist for wave speeds $c \geq c^*$ when $R_0> 1$, with $c^*$ denoting the minimal wave speed. Conversely, no traveling waves are observed for $c

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems.

This paper investigates the speed selection mechanism for traveling wave fronts of a reaction-diffusion-advection lattice stream-population model with the Allee effect. First, the asymptotic behaviors of the traveling wave solutions are given. Then, sufficient conditions for the speed determinacy of the traveling wave are successfully obtained by constructing appropriate upper and lower solutions. We examine the model with the reaction term $ f (\psi) = \psi(1-\psi)(1+\rho\psi) $, with $ \rho $ being a nonnegative constant, as a specific example. We give a novel conjecture that there exists a critical value $ \rho_c > 1 $, such that the minimal wave speed is linearly selected if and only if $ \rho\leq\rho_c $. Finally, our speculation is verified by numerical calculations.

This paper investigates a nonlocal dispersal epidemic model under the multiple nonlocal distributed delays and nonlinear incidence effects. First, the minimal wave speed c ∗ and the basic reproduction number R 0 are defined, which determine the existence of traveling wave solutions. Second, with the help of the upper and lower solutions, Schauder's fixed point theorem, and limiting techniques, the traveling waves satisfying some asymptotic boundary conditions are discussed. Specifically, when ℛ 0 > 1 , for every speed c > c ∗ there exists a traveling wave solution satisfying the boundary conditions, and there is no such traveling wave solution for any 0 < c < c ∗ when ℛ 0 > 1 or c > 0 when ℛ 0 < 1. Finally, we analyze the effects of nonlocal time delay on the minimum wave speed. [ABSTRACT FROM AUTHOR]

The intricate relationship between prestimulus alpha oscillations and visual contrast detection variability has been the focus of numerous studies. However, the causal impact of prestimulus alpha traveling waves on visual contrast detection remains largely unexplored. In our research, we sought to discern the causal link between prestimulus alpha traveling waves and visual contrast detection across different levels of mental fatigue. Using electroencephalography alongside a visual detection task with 30 healthy adults (13 females; 17 males), we identified a robust negative correlation between prestimulus alpha forward traveling waves (FTWs) and visual contrast threshold (VCT). Inspired by this correlation, we utilized 45/-45° phase-shifted transcranial alternating current stimulation (tACS) in a sham-controlled, double-blind, within-subject experiment with 33 healthy adults (23 females; 10 males) to directly modulate these alpha traveling waves. After the application of 45° phase-shifted tACS, we observed a substantial decrease in FTW and an increase in backward traveling waves, along with a concurrent increase in VCT, compared with the sham condition. These changes were particularly pronounced under a low fatigue state. The findings of state-dependent tACS effects reveal the potential causal role of prestimulus alpha traveling waves in visual contrast detection. Moreover, our study highlights the potential of 45/-45° phase-shifted tACS in cognitive modulation and therapeutic applications. [ABSTRACT FROM AUTHOR]

Many pathogens evolve to escape immunity, yet it remains difficult to predict whether immune pressure will lead to diversification, serial replacement of one variant by another, or more complex patterns. Pathogen strain dynamics are mediated by cross-protective immunity, whereby exposure to one strain partially protects against infection by antigenically diverged strains. There is growing evidence that this protection is influenced by early exposures, a phenomenon referred to as original antigenic sin (OAS) or imprinting. In this paper, we derive constraints on the emergence of the pattern of successive strain replacements demonstrated by influenza, SARS-CoV-2, seasonal coronaviruses, and other pathogens. We find that OAS implies that the limited diversity found with successive strain replacement can only be maintained if R0 is less than a threshold set by the characteristic antigenic distances for crossprotection and for the creation of new immune memory. This bound implies a "speed limit" on the evolution of new strains and a minimum variance of the distribution of infecting strains in antigenic space at any time. To carry out this analysis, we develop a theoretical model of pathogen evolution in antigenic space that implements OAS by decoupling the antigenic distances required for protection from infection and strainspecific memory creation. Our results demonstrate that OAS can play an integral role in the emergence of strain structure from host immune dynamics, preventing highly transmissible pathogens from maintaining serial strain replacement without diversification. [ABSTRACT FROM AUTHOR]

One of the significant causes of death globally is cancer (http://www.who.org/). Another critical problem is obesity, which is associated with an increased cancer threat. This work provides insight into how obesity contributes to cancer progression and metastasis. We developed a diffusive obesity-cancer model consisting of cancer cells, normal cells, fat cells, macrophages, and an extracellular matrix (ECM) for this aim. We have directed the formed model's global existence and non-negativity. Equilibrium points for the related ODE are calculated, and its existence and stability study is also done. We present a traveling wave analysis of the obesity-cancer model and have calculated the minimum wave speed. Using a combination of analytic and numerical results of traveling waves, we conjecture that the minimal wave speed depends on fat cells' diffusive rate and haptotaxis coefficient. We followed the theory of the Partial Rank Correlation Coefficient (PRCC) to carry out a global sensitivity analysis to evaluate the most sensitive parameters reliable for cancer progression. We delivered a comprehensive numerical analysis of our deterministic and diffusive models (in 1D and 2D) and analogized the result. Numerical simulation of corresponding spatially explicit systems conveys complex spatio-temporal dynamics, resulting in the appearance of patterns. It also discloses that cancer spread increases with increased haptotaxis coefficient and growth rate of obese cells. Our simulation confirms that the degradation of the ECM increases cancer spread and density. [ABSTRACT FROM AUTHOR]

In this paper, the coupled system nonlinear (2 + 1)-dim Broer–Kaup–Kupershmit (BKK) equation under consideration is based on extension of modified rational expansion method. The various kinds of multiple soliton solutions named anti-kink soliton, traveling wave solutions, periodic solution, dark soliton, kink soliton, bright soliton, anti-kink bright solitons, kink bright solitons for nonlinear BKK system are successfully constructed under symbolic computation Mathematica. The calculated results are very interesting, different and novel which have not been investigated in past literature. The graphical demonstration of constructed solutions show by contour, 2D and 3D shape by using computer software Mathematica. The constructed multiple soliton solutions prove that EMRE approach is very straightforward, efficient, reliable and powerful for the investigation of other nonlinear partial differential equations. The investigated results will be described and play a keen role in the study of nonlinear physical phenomena in the area of physics and engineering. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of spatio-temporal dynamics for a time-space periodic population growth model with impulsive birth. We first formulate a discrete-time semiflow by the 1-year solution map, and obtain a threshold-type result for the semiflow with spatially periodic initial data. Then we establish the existence and the computational formulas of the spreading speeds and prove the coincidence of the spreading speeds with the minimal speeds of spatially periodic traveling waves in the monotone case. Further, we investigate the global dynamics of this model in a bounded spatial domain. Finally, we conduct numerical simulations to verify our analytical results and illustrate some interesting findings. [ABSTRACT FROM AUTHOR]

We examine the vanishing adsorption limit of solutions of Riemann problems for the Glimm–Isaacson model of chemical flooding of a petroleum reservoir. A contact discontinuity is deemed admissible if it is the limit of traveling waves or rarefaction waves for an augmented system that accounts for weak chemical adsorption onto the rock. We prove that this criterion justifies the admissibility criteria adopted previously by Keyfitz–Kranzer, Isaacson–Temple and de Souza–Marchesin, provided that the fractional flow function depends monotonically on chemical concentration. We also demonstrate that the adsorption criterion selects the undercompressive contact discontinuities required to solve the general Riemann problem in an example model with non-monotone dependence. [ABSTRACT FROM AUTHOR]

The existence of weak and strong detonation waves for a version of the Majda's model is proved. In fact, the existence of traveling wave solutions for an exothermic combustion involving more than one reaction is studied. [ABSTRACT FROM AUTHOR]

We study the asymptotic stability of a front connecting two unstable states. Such a structure typically appears when the stable state behind a Fisher--Kolmogorov--Petrovskii--Piskunov front destabilizes when going through an essential Turing bifurcation, giving rise to oscillating patterns. Despite the instability of both end-states, we obtain for the first time stability of such a structure against suitably localized perturbations, with algebraic temporal decay t-3/2. To deal with the instability behind the front, we simultaneously control the error in two different norms. In the first norm, enhanced diffusive decay is obtained at a linear level through pointwise resolvent estimates. In the second norm, better suited for nonlinear analysis, we show that the error stays bounded in time by use of mode filters. [ABSTRACT FROM AUTHOR]

In this paper, we apply a machine-learning approach to learn traveling solitary waves across various physical systems that are described by families of partial differential equations (PDEs). Our approach integrates a novel interpretable neural network (NN) architecture, called Separable Gaussian Neural Networks (SGNN) into the framework of Physics-Informed Neural Networks (PINNs). Unlike the traditional PINNs that treat spatial and temporal data as independent inputs, the present method leverages wave characteristics to transform data into the so-called co-traveling wave frame. This reformulation effectively addresses the issue of propagation failure in PINNs when applied to large computational domains. Here, the SGNN architecture demonstrates robust approximation capabilities for single-peakon, multi-peakon, and stationary solutions (known as "leftons") within the (1+1)-dimensional, b-family of PDEs. In addition, we expand our investigations, and explore not only peakon solutions in the a b -family but also compacton solutions in (2+1)-dimensional, Rosenau-Hyman family of PDEs. A comparative analysis with multi-layer perceptron (MLP) reveals that SGNN achieves comparable accuracy with fewer than a tenth of the neurons, underscoring its efficiency and potential for broader application in solving complex nonlinear PDEs. [ABSTRACT FROM AUTHOR]

Generalist predation generally stabilizes the prey–predator dynamics since a generalist predator utilizes a variety of food sources to survive and shows prey-switching behavior at low focal prey density by reducing the predation pressure. On the other hand, the presence of Allee effect can potentially lead to a fairly complex prey–predator dynamics including the suppression of "the paradox of enrichment". In this paper, we explore the combined influence of reproductive Allee effect in prey growth and generalist predation on the resulting temporal as well as spatio-temporal dynamics. The temporal model mainly exhibits bistability in terms of stable equilibria. For the corresponding spatio-temporal model, we perform detailed theoretical analysis regarding the non-existence and existence of spatially heterogeneous steady states, and also provide conditions for Turing instability. The spatio-temporal model mainly exhibits stationary patterns and traveling wave solutions apart from the usual spatially homogeneous solutions. Our study reveals that oscillatory dynamics is less probable in the presence of abundant alternative food sources to predator population. [ABSTRACT FROM AUTHOR]

In this paper, we consider the steadiness of symmetric solutions to two dispersive models in shallow water and hyperelastic mechanics, respectively. These models are derived previously in the two-dimensional setting and can be viewed as the generalization of the Camassa–Holm and Kadomtsev–Petviashvili equations. For these two models, we prove that the symmetry of classical solutions implies steadiness in the horizontal direction. We also confirm the connection between symmetry and steadiness for solutions in weak formulation, which covers in particular the peaked solutions. [ABSTRACT FROM AUTHOR]

Traveling waves of neural activity emerge in cortical networks both spontaneously and in response to stimuli. The spatiotemporal structure of waves can indicate the information they encode and the physiological processes that sustain them. Here, we investigate the stimulus-response relationships of traveling waves emerging in adaptive neural fields as a model of visual motion processing. Neural field equations model the activity of cortical tissue as a continuum excitable medium, and adaptive processes provide negative feedback, generating localized activity patterns. Synaptic connectivity in our model is described by an integral kernel that weakens dynamically due to activity-dependent synaptic depression, leading to marginally stable traveling fronts (with attenuated backs) or pulses of a fixed speed. Our analysis quantifies how weak stimuli shift the relative position of these waves over time, characterized by a wave response function we obtain perturbatively. Persistent and continuously visible stimuli model moving visual objects. Intermittent flashes that hop across visual space can produce the experience of smooth apparent visual motion. Entrainment of waves to both kinds of moving stimuli are well characterized by our theory and numerical simulations, providing a mechanistic description of the perception of visual motion. [ABSTRACT FROM AUTHOR]