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We consider a class of cooperative reaction-diffusion systems with free boundaries in one space dimension, where the diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and they are allowed not to appear in some of the equations in the system. Such a system covers various models arising from mathematical biology, in particular a West Nile virus model and an epidemic model considered recently in [16] and [44] , respectively, where a “spreading-vanishing” dichotomy is known to govern the long time dynamical behaviour, but the question on spreading speed was left open. In this paper, we develop a systematic approach to determine the spreading profile of the system, and obtain threshold conditions on the kernel functions which decide exactly when the spreading has finite speed, or infinite speed (accelerated spreading). This relies on a rather complete understanding of both the associated semi-waves and travelling waves. When the spreading speed is finite, we show that the speed is determined by a particular semi-wave. This is Part 1 of a two part series. In Part 2, for some typical classes of kernel functions, we will obtain sharp estimates of the spreading rate for both the finite speed case, and the infinite speed case.

Many systems in life sciences have been modeled by reaction–diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism like impulsive reaction–diffusion equations is necessary to analyze them. While several works tackled the issue of traveling waves for monotone reaction–diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction–diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reaction–diffusion equations. Our first result deals with the existence of traveling waves for monotone systems of impulsive reaction–diffusion equations. Our second result tackles the computation of spreading speeds for monotone systems of impulsive reaction–diffusion equations. We apply our methodology to a planar system of impulsive reaction–diffusion equations that models tree–grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion.

In this paper, the new travelling wave solutions of the (2+1)-dimensional extended Calogero-Bogoyavlenskii-Schiff (ECBS) equation are investigated. The main aim of this work is to find the new exact solutions with the aid of relatively new ( G ′ G ′ + G + A ) -expansion method. Moreover, the physical interpretation of the nonlinear phenomena is reported through the exact solutions, which indicate the efficacy of the proposed method. Furthermore, the recovered solutions are periodic and solitary wave solutions which are presented graphically.

These notes are dedicated to the analysis of the one-dimensional free-congested Navier–Stokes equations. After a brief synthesis of the results obtained in Dalibard and Perrin (Commun Math Sci 18(7):1775–1813, 2020) related to the existence and the asymptotic stability of partially congested profiles associated to the soft congestion Navier-Stokes system, we present a first local well-posedness result for the one-dimensional free-congested Navier–Stokes equations.

Silicon Mach-Zehnder modulator (SMZM)is fit to high-contrast optical modulation in wide ranges ofspectral along 50 Gbaud symbol rates. Some times itsefficiency is lower in wavelength about 1.31 μm than 1.55μm, it can decrease the Phase Shifter (PS) efficiency andoccupies large amount of data rates. In this manuscript,the plus-shaped PN junction Phase Shifter (PS) isproposed. The major aim of the proposed work is to createan optimum CD type plus-shaped PS including lesser VπL.Silicon MZM including proposed PS can satisfy thehigher-speed data transmission requirement on theapplications of inter with intra data centre. The objectiveof this method is to increase the modulation efficiency(ME) by decreasing the optical loss for higher-speed datarate (DR). To get greater efficiency of modulation, the Pdoped region width and the thickness of doped regions arediffer under slabs. The simulation analysis of circuit-levelis executed in the proposed PS acquired at travelling waveelectrode (TWE) silicon Mach Zehnder modulator. In 80Gbps, the 12.39 dB maximal extinction ratio along8.67×10-8 bit error rate (BER) was acquired in VπLπ of1.05 V.cm for 3.5 mm PS length. The measured intrinsicbandwidth of 3 dB denotes ~38 GHz, whereas thetransmission energy per bit denotes 1.71pJ/bit. Moreexamines are carried out to recognize the maximalcommunication distance using proposed PS under SMZMfor the requirements of data centre.

We consider the third order Benjamin-Ono equation on the torus ∂ t u = ∂ x ( − ∂ x x u − 3 2 u H ∂ x u − 3 2 H ( u ∂ x u ) + u 3 ) . We prove that for any t ∈ R , the flow map continuously extends to H r , 0 s ( T ) if s ≥ 0 , but does not admit a continuous extension to H r , 0 − s ( T ) if 0 s 1 2 . Moreover, we show that the extension is weakly sequentially continuous in H r , 0 s ( T ) if s > 0 , but is not weakly sequentially continuous in L r , 0 2 ( T ) . We then classify the traveling wave solutions for the third order Benjamin-Ono equation in L r , 0 2 ( T ) and study their orbital stability.

We consider a coupled reaction-advection-diffusion system based on the Fisher-KPP and Burgers equations. These equations serve as a one-dimensional version of a model for a reacting fluid in which the arising density differences induce a buoyancy force advecting the fluid. We study front propagation in this system through the lens of traveling waves solutions. We are able to show two quite different behaviors depending on whether the coupling constant $\rho$ is large or small. First, it is proved that there is a threshold $\rho_0$ under which the advection has no effect on the speed of traveling waves (although the advection is nonzero). Second, when $\rho$ is large, wave speeds must be at least $\mathcal{O}(\rho^{1/3})$. These results together give that there is a transition from pulled to pushed waves as $\rho$ increases. Because of the complex dynamics involved in this and similar models, this is one of the first precise results in the literature on the effect of the coupling on the traveling wave solution. We use a mix of ordinary and partial differential equation methods in our analytical treatment, and we supplement this with a numerical treatment featuring newly created methods to understand the behavior of the wave speeds. Finally, various conjectures and open problems are formulated., Comment: 33 pages, 2 figures

In this paper, we are concerned with the existence of traveling waves in a class of delayed higher dimensional lattice differential systems with competitive interactions. Due to the lack of quasimonotonicity for reaction terms, we use the cross iterative and Schauder’s fixed-point theorem to prove the existence of traveling wave solutions. We apply our results to delayed higher-dimensional lattice reaction-diffusion competitive system.

In present paper, multiple rogue wave solutions of (2+1)-dimensional Yu–Toda–Sasa–Fukuyama equation were studied by applying the traveling wave transformation and the Hirota bilinear method. We obt...

This paper studies the minimal wave speed of travelling wave solutions in a predator–prey integrodifference system. Even if the predator vanishes, the corresponding scalar equation of prey may be n...

We investigate the traveling wave solutions of a three-species system involving a single predator and a pair of strong-weak competing preys. Our results show how the predation may affect this dynamics. More precisely, we describe several situations where the environment is initially inhabited by the predator and by either one of the two preys. When the weak competing prey is an aboriginal species, we show that there exist traveling waves where the strong prey invades the environment and either replaces its weak counterpart, or more surprisingly the three species eventually co-exist. Furthermore, depending on the parameters, we can also construct traveling waves where the weaker prey actually invades the environment initially inhabited by its strong competitor and the predator. Finally, our results on the existence of traveling waves are sharp, in the sense that we find the minimal wave speed in all those situations., Comment: Journal of Differential Equations, Elsevier, 2021

We have utilized three novel methods, called generalized direct algebraic, modified F-expansion and improved simple equation methods to construct traveling wave solutions of the system of shallow water wave equations and modified Benjamin-Bona-Mahony equation. After substituting particular values of the parameters, different solitary wave solutions are derived from the exact traveling wave solutions. It is shown that these employed methods are more powerful tools for nonlinear wave equations.

In this paper, we investigate the KdV equation with dual-power law nonlinearity. As a result, we have obtained general exact travelling wave soliton solutions such as bright soliton solution, dark soliton solution and periodic solution. These solutions have many free parameters in such away that they may be used to simulate many experimental situations. The main contribution in this work is to give the general solution of the obtained equations with different values of parameters $$n$$ .

We establish the existence of traveling waves for a Lotka-Volterra competition-diffusion model with a shifting habitat. It is assumed that the growth rate of each species is nondecreasing along the x-axis, positive near ∞ and negative near −∞, and shifting rightward at a speed c. We show that under appropriate conditions, for the case that one species is competitively stronger near ∞ and the case that both species coexist near ∞, there exists a critical number c ¯ ( ∞ ) such that for any c > c ¯ ( ∞ ) there exists a forced traveling wave with speed c connecting the origin and a semi-trivial steady state and for c c ¯ ( ∞ ) such a traveling wave does not exist. We also show that when a coexistence steady state exists, for any c > 0 , there is a forced traveling wave with speed c connecting the origin and the coexistence steady state.

The electrical curtain board (ECB) is constituted of parallel linear electrodes deposited on a dielectric layer plate and supplied by a poly-phase electrical voltage. This paper reports a study of ...

In this paper, we study the minimal speed (spreading speed) selection mechanism of traveling waves to a periodic diffusive Lotka-Volterra model with monostable nonlinearity. The method of upper and lower solutions pair is applied to establish the existence of traveling waves. We prove that the nature of nonlinear selection is to find a lower solution pair with the first species decaying in a faster rate. By novel constructions of various upper and lower solutions, we obtain a number of new results on the minimal speed determinacy. Numerical simulations are carried out to illustrate all of our discoveries.

Over the last few years, a direct method known as the new auxiliary method has been shown to solve non-linear partial differential equations effectively where this method can give more exact solutions compared to the traditional direct methods. Due to these promising results, efforts are now being made to extend this method in solving the more complicated non-linear fractional partial differential equations. The new auxiliary method is a very powerful, felicitous, and effective method to get exact and traveling wave solutions of partial differential equations. Therefore, we apply the new auxiliary method to solve for the traveling wave solutions of the space-time fractional Cahn-Hilliard equation and the space-time fractional symmetric regularized long-wave equation. Furthermore, it is considered as more general method for solving that time of equations because it is shown that more exact solutions for the traveling wave on these equations will be obtained with the application of the new auxiliary method.

In this article, the Nonlinear Schrodinger (NLS) equation, which is very important in physics, have been discussed. The analytical solution of this equation with the (1/G′)-expansion method and its...

In this paper, we obtain new soliton solutions of one of the most important equations in biology (fractional time coupled nerve fibers) using two algorithm schemes, namely, exp − ψ ξ expansion function method and θ ′ ξ / θ 2 ξ expansion methods. The equation and the solution methods have free parameters which help to make the obtained solutions are dynamics and more readable for dealing with fractional parameter and the initial and boundary value problem. As a result, various new exact soliton solutions for the considered model are derived which include the hyperbolic, rational, and trigonometric functions, and other solutions are obtained. In addition, the obtained results proved that the used methods give better performance compared with existing methods in the literature.

The use of improved generalized Riccati equation mapping method has been demonstrated to find some new exact travelling wave solutions to space-time fractional non-liner double dispersive equation (DDE). The equation is used for modelling wave propagation in elastic inhomogeneous Murnaghan's rod. We have used Caputo's fractional derivative to achieve the fractional version of Murnaghan's rod equation. Improved generalized Riccati equation mapping method proves to be very effective tool to find a variety of soliton solutions. As a result, we found many new and general solutions including dark, combined dark-bright, singular periodic wave, combined singular periodic wave solutions and rational solutions. We have simulated the solitons, to check their types, with the help of graphs and all the solutions obtained in this article have been verified by back substitution in original equation by using Maple 17.

In this paper, by using the dynamical system theory and simulation method, we study a generalized Broer-Kaup (gBK) equation. Under different parameter conditions, the travelling wave solutions such as kink wave, periodic blow-up wave, periodic wave and solitary wave are given, and the dynamic characters of these solutions are investigated.

This work is concerned with the dynamical behaviors of a singular predator-prey model. We first review some well-known results obtained recently. Then we give some new results on the spreading speed of the predator, the existence vs non-existence of traveling waves connecting the predator-free state to the co-existence state, and the existence vs non-existence of spatially periodic traveling waves to this singular predator-prey system.

Gui Ling Ji (GLJ), an ancient reputable traditional Chinese medicine (TCM) formula prescription, has been applied for the treatment of oligospermia and asthenospermia in clinical practice. However, its inherent compounds have not yet been systematically elucidated, which hampers developing standards or guidelines for quality evaluation and even the understanding of pharmacological effects. In this study, an integrated approach has been established for comprehensive structural characterization of GLJ. Mass spectrometry datasets of GLJ and each of the single herb medicines in this prescription have been developed by dynamic exclusion fast data-dependent acquisition and high-definition data-independent acquisition modes on ultra-high-performance liquid chromatography coupled with travelling wave ion mobility quadrupole time-of-flight mass spectrometry (UPLC-TWIMS-QTOF-MS). A global natural product social molecular networking (GNPS) platform was then applied for the visualization of chemical space of GLJ and further for the high throughput identification of the targeted or untargeted compounds due to the support of data-transmitting from each single herbal medicine to the formula GLJ. Moreover, drift time, predicted CCS, and diagnostic fragment ions were induced for annotating isomer compounds. Consequently, based on molecular network and library hits, a total of 257 compounds from GLJ, which were classified into 4 structural types, were positively or tentatively characterized. Among them, 20 potential new compounds were detected and 30 pairs of isomers were comprehensively distinguished. The established strategy was effective for attribution, classification, recognition of various constituents, and also was valuable for integrating large amounts of disordered MS/MS data and mining trace compounds in other complex chemical or biochemical systems.

In this paper, we first establish the existence of semi-traveling wave solutions to a diffusive generalized Holling-Tanner predator-prey model in which the functional response may depend on both the predator and prey populations. Then, by constructing the Lyapunov function, we apply the obtained result to show the existence of traveling wave solutions to the diffusive Holling-Tanner predator-prey models with various functional responses, including the Lotka-Volterra type functional response, the Holling type Ⅱ functional response and the Beddington-DeAngelis functional response.

We complete the description, initiated in [ 6 ], of a free boundary traveling at constant speed in a half plane, where the propagation is controlled by a line having a large diffusion on its own. The main result of this work is that the free boundary is asymptotic to a line at infinity, whose angle to the horizontal is dictated by the velocity of the wave on the line. This helps understanding some counter-intuitive simulations of [ 7 ].

Autonomous active, elastic filaments that interact with each other to achieve cooperation and synchrony underlie many critical functions in biology. The mechanisms underlying this collective response and the essential ingredients for stable synchronization remain a mystery. Inspired by how these biological entities integrate elasticity with molecular motor activity to generate sustained oscillations, a number of synthetic active filament systems have been developed that mimic oscillations of these biological active filaments. Here, we describe the collective dynamics and stable spatiotemporal patterns that emerge in such biomimetic multi-filament arrays, under conditions where steric interactions may impact or dominate the collective dynamics. To focus on the role of steric interactions, we study the system using Brownian dynamics, without considering long-ranged hydrodynamic interactions. The simulations treat each filament as a connected chain of self-propelling colloids. We demonstrate that short-range steric inter-filament interactions and filament roughness are sufficient - even in the absence of inter-filament hydrodynamic interactions - to generate a rich variety of collective spatiotemporal oscillatory, traveling and static patterns. We first analyze the collective dynamics of two- and three-filament clusters and identify parameter ranges in which steric interactions lead to synchronized oscillations and strongly occluded states. Generalizing these results to large one-dimensional arrays, we find rich emergent behaviors, including traveling metachronal waves, and modulated wavetrains that are controlled by the interplay between the array geometry, filament activity, and filament elasticity. Interestingly, the existence of metachronal waves is non-monotonic with respect to the inter-filament spacing. We also find that the degree of filament roughness significantly affects the dynamics - specifically, filament roughness generates a locking-mechanism that transforms traveling wave patterns into statically stuck and jammed configurations. Taken together, simulations suggest that short-ranged steric inter-filament interactions could combine with complementary hydrodynamic interactions to control the development and regulation of oscillatory collective patterns. Furthermore, roughness and steric interactions may be critical to the development of jammed spatially periodic states; a spatiotemporal feature not observed in purely hydrodynamically interacting systems.

The article deals with the Fermi–Pasta–Ulam–type systems that describe infinite systems of particles on a 2D lattice. The main result concerns the existence of the solutions corresponding to traveling waves with periodic and vanishing profiles. By means of the critical point theory, the sufficient conditions for the existence of such solutions are obtained.

This paper presents a multi-parameter model of an ultrasonic motor for speed control. Firstly, the model is divided into the steady-state part and the dynamic part, and then the speed mapping relationship of amplitude, frequency, phase difference is described by linear function, exponential function, and hyperbolic tangent function, respectively in steady-state modeling. The least-square transfer function identification is used in the dynamic velocity modeling. This paper also constructs a modeling flowchart from the discrete models to the state-space function based on the pairwise combination of parameters, which can realize the flexible control of different parameters. Finally, the error box diagram of the model and the comparison between the measured results and simulation ones both verify the effectiveness of the model.

Better understanding of the characteristics of the traveling wave and three-dimensional trajectory related to motion on the surface of the stator is very important for the design and performance improvement of the ultrasonic motors. In this paper, an accurate finite element model of a single stator with a fully coupled piezoelectric layer was established at a moderate computational cost. The finite element model was verified by experimental test at the inverse resonance point. Based on this model, the traveling wave and three-dimensional trajectory of stator surface, including the influence of the input voltage on the phase and amplitude of the displacements in three directions, are investigated. The results show that the trajectory of particles on the stator surface is an ellipse in three-dimensional space due to the phase differences between the three components of displacement in the radial, circumferential and axial directions. The amplitude of radial displacement is about 39.5% of that in the circumferential displacement, which should not be neglected.

In this paper, the classification of all single traveling wave solutions to generalized fractional Gardner equations is presented by utilizing the complete discrimination system method. Under the fractional traveling wave transformation, generalized fractional Gardner equations can be reduced to an ordinary differential equations. All possible exact traveling wave solutions are given through the complete discrimination system of the fourth-order polynomial. Moreover, graphical representations of different kinds of the exact solutions reveal that the method is of significance for searching the exact solutions to generalized fractional Gardner equations.

We investigate the connection between the existence of an explicit travelling wave solution and the travelling wave with minimal speed in a scalar monostable reaction-diffusion equation.

As a new type of refrigeration technology, the traveling-wave thermoacoustic refrigerator offers advantages that include high efficiency, reliability and environmental friendliness. To date, because of problems such as low power utilization and high power recovery losses, traveling-wave thermoacoustic refrigerators for use in room temperature applications have not been widely studied. In this paper, following an investigation of the traditional single-stage traveling-wave thermoacoustic refrigerator, a multi-stage traveling-wave thermoacoustic refrigerator is proposed and the working mechanism of this refrigerator is studied numerically using SAGE software. The calculation results show that the proposed multi-stage traveling-wave thermoacoustic refrigerator can enhance the utilization of the input acoustic work effectively, thereby improving the cooling power of the refrigerator with high cooling efficiency. As a result, the cooling power increases from 2.17 kW for a single-stage refrigerator to 6.42 kW for a seven-stage refrigerator, while the acoustic work utilization rate increases from 0.26 to 0.82, and the coefficient of performance changes from 2.60 to 3.19. The calculation results also indicate that three to five stages may be most suitable for the multi-stage traveling-wave thermoacoustic refrigerator when working within the temperature range of interest here by striking a balance between cooling efficiency and cooling power.

We use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion–reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of interest. We determine the minimum wave speed, $$c^*$$ c ∗ , and investigate its relation to the spectral stability of a desingularised linear operator associated with the travelling wave solutions.

The extended complex method is investigated for exact analytical solutions of nonlinear fractional Liouville equation. Based on the work of Yuan et al., the new rational, periodic, and elliptic function solutions have been obtained. By adjusting the arbitrary values to the constants in the constructed solutions, it can describe the physical phenomena to the traveling wave solutions, since traveling wave has significant value in applied sciences and engineering. Our results indicate that the extended complex technique is direct and easily applicable to solve the nonlinear fractional partial differential equations (NLFPDEs).

Applications of a new function introduced by Kudryashov [Optik. 2020;206:163550] to obtain solitary wave solutions of nonlinear PDEs through their travelling wave reductions are considered. The Kud...

The main aim of this work is to investigate numerical solutions of the two different types of the fifth-order modified Kawahara equation namely bell-shaped soliton solutions and travelling wave solutions that occur thereby the different type of the Korteweg–de Vries equation. For this approach, we have used an effective and simple type of finite difference method namely Crank-Nicolson scheme for time integration and third-order modified cubic B-spline-based differential quadrature method for space integration. We preferred the third-order modified cubic B-splines to solve the fifth-order partial differential equation because of by using low energy, less algebraic process and produce better results than earlier works. To display the efficiency and accuracy of the present fresh approach famous test problems namely bell-shaped single soliton that has negative amplitude and travelling wave solutions that have the both of the positive and negative amplitudes are solved and the error norms L 2 and L ∞ are calculated and compared with earlier works. Comparison of the error norms show that present fresh approach obtained superior results than earlier works by using same parameters. At the same time, two lowest invariants of the test problems during the simulations are calculated and reported. Besides those, relative changes of invariants are computed and reported.

We consider a population dynamics model with degenerate diffusion and time delay. We discover sharp-oscillatory waves with sharp edges and non-decaying oscillations arising from density-dependent dispersal and reproduction with age structure. Degenerate diffusion and bad effect of time delay prevent the use of existing approaches. Here, we develop a new delayed iteration framework to show the existence of these peculiar waves. In particular, the estimate of the admissible wave speed is highly nontrivial. We employ a new phase transform method coupled with the detailed analysis of phase energy. Furthermore, we give a complete characterization of the dynamical behaviors of various kinds of waves. Our results indicate that simple invasion rules can generate complex wave patterns and provide some interesting insights into the ecological dynamics.

We study spreading phenomena in a two-component reaction-diffusion system, modeling the Neolithic transition from hunter-gatherer societies to agricultural societies in Europe. When farmer populations were introduced into the region inhabited by hunter-gatherer populations, farmer populations spread as an expanding wave through Europe. The speed of the wave is observed to tend to be a constant. We analytically compute the spreading velocity of the model, and characterize the longtime behavior of solutions of the model. Without comparison principle, we introduce a new technique that can locate the spreading front in reaction-diffusion systems.

The electric component of the RF field at the input of the traveling wave accelerating structure (operating frequency of 2856 MHz) of the LUE-200 linear electron accelerator – a driver for the pulsed source of the IREN facility developed at the Frank Laboratory of Neutron Physics of the Joint Institute for Nuclear Research (Dubna, Russia) – is considered. The estimates of the focusing effect of the RF field on the nonrelativistic electron beam injected into the structure after the buncher performed for the thin electric lens model have shown that the focusing effect of the edge RF field for the beam particles turns out to be sufficiently rigid.

We improve the decay argument by Bona and Li (1997) [5] for solitary waves of general dispersive equations and illustrate it in the proof for the exponential decay of solitary waves to steady Degasperis-Procesi equation in the nonlocal formulation. In addition, we give a method which confirms the symmetry of solitary waves, including those of the maximum height. Finally, we discover how the symmetric structure is connected to the steady structure of solutions to the Degasperis-Procesi equation, and give a more intuitive proof for symmetric solutions to be traveling waves. The improved argument and new method above can be used for the decay rate of solitary waves to many other dispersive equations and will give new perspectives on symmetric solutions for general evolution equations.

In this paper, the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili(BKP) equation is studied applying Lie symmetry analysis. We apply the Lie symmetry method to the (3 + 1)-dimensional generalized BKP equation and derive its symmetry reductions. Based on these symmetry reductions, some exact traveling wave solutions are obtained by using the tanh method and Kudryashov method. Finally, the conservation laws to the (3 + 1)-dimensional generalized BKP equation are presented by invoking the multiplier method.

A hydrodynamic-type system taking into account nonlocal effects is investigated. The exact traveling wave solutions including smooth solitary waves solutions, pseudo-peakons, periodic peakons, compactons, kink, and anti-kink wave solutions and so on are derived via the method of dynamical systems and the theory of singular traveling wave systems. It is worth pointing out that the uncountably infinitely many solitary wave solutions, kink, and anti-kink solutions are new solutions for the modeling system.

Nonlinear partial differential equations (NLPDEs) are an inevitable mathematical tool to explore a large variety of engineering and physical phenomena. Due to this importance, many mathematical approaches have been established to seek their traveling wave solutions. In this study, the researchers examine the Gardner equation via two well-known analytical approaches, namely, the improved tanΘϑ-expansion method and the wave ansatz method. We derive the exact bright, dark, singular, and W-shaped soliton solutions of the Gardner equation. One can see that the methods are relatively easy and efficient to use. To better understand the characteristics of the theoretical results, several numerical simulations are carried out.

In this paper, we consider the existence of traveling wave fronts in a Belousov-Zhabotinskii system with delay. By traveling wave transformation and time scale transformation, we change the Belousov-Zhabotinskii system with delay into a singularly perturbed differential system. By applying geometric singular perturbation theory, we construct a locally invariant manifold for the associated traveling wave equation and obtain the traveling wave fronts for the equation by using the Fredholm orthogonality. Finally, we discuss the asymptotic behaviors of traveling wave solutions by applying the asymptotic theory.

In a recent work [1] , a mathematical model is derived to explore the role that the white-tail deer plays in the geographic spread of the black-legged tick Ixodes scapularis in the northeast of the United States of America. The threshold dynamics is rigorously investigated in terms of the basic reproduction number, for the cases of the 1-D whole space Ω = R and general bounded spatial domain Ω with homogeneous Neumann and Direchlet boundary conditions. However, the minimal wave speed and spread speed of the model, which are the motivation of this model and thus most important, are only explored numerically. In the present paper, we offer a rigorous theoretical confirmation of what are numerically observed in [1] , concluding that if the basic reproduction number is larger than one, the model allows a spread speed which is also the minimal speed of traveling wave fronts, and this speed is linearly deterministic.

Using symbolic calculations, a method of local asymptotics is developed, which describes the diffraction focusing of electromagnetic and acoustic fields for edge and corner catastrophes. Explicit expressions are found for the unfolding coefficients, the functional module, and the phase of the traveling wave, when the family of primary (geometrical-optical) and secondary (edge) rays form focuses on the cuspoid $${\mathrm{\boldsymbol A}}_{\mathrm{2}}$$ type (simple edge catastrophe $${\mathrm{\boldsymbol F}}_{\mathrm{4}}$$ ), when the family of primary and secondary rays form focuses on the $${\mathrm{\boldsymbol A}}_{\mathrm{3}}$$ type (unimodal edge catastrophe $${\mathrm{\boldsymbol K}}_{\mathrm{4,2}}$$ ), as well as for a corner catastrophe $${\mathrm{\boldsymbol A}_{1}}^{4}$$ .

In this work, the new analytical exact solution of a highly nonlinear fractional partial differential equation (FPDE) has been presented; specifically space-time fractional (3+1)-dimensional Jimbo–Miwa (JM) equation. As a consequence of the applied extended (G'/G)-expansion method, the new analytical exact solution of the governing equation has been acquired successfully. Moreover, the physical natures of the solutions have been analyzed here by means of numerical simulations.

The means of measuring humidity based on the use of the ultrahigh frequency method have been recently gaining widespread use, because of its simple, robust construction and high measuring accuracy. We used the advanced waveguide ultrahigh frequency method of measuring the moisture content of natural gas which, in contrast to the known the use of a traveling wave in a waveguide, is proposed. In this case, the interaction with waves of the ultrahigh frequency range changes the dielectric properties of the gas, and this change is registered. On the basis of an improved ultrahigh frequency method of humidity measurement, a device for natural gas humidity control using a traveling wave in a waveguide is proposed. The investigations have shown that a comparative channel increased the measurement accuracy, as a two-channel system – in contrast to a single-channel – eliminates the instability of the value of the input signal supplied to the generator. The principle of operation of a natural gas humidity control device that contains an ultrahigh frequency generator, attenuators, waveguide tees, a waveguide section for comparison, temperature sensor and pressure switches for the comparative and measuring channels, a measuring cuvette, amplifier, microprocessor, and display unit is described. A mathematical model of a natural gas humidity control device, which takes into account the values of the dielectric permittivity of the measuring gas and reference channels and contains correction factors for temperature, the use of which increases the accuracy of humidity measurement, is proposed. The lower and upper calibration points of the natural gas humidity control device are defined. The influence of correction factors for the temperature at the measurement error of the humidity is analyzed.

In this paper, we have obtained the traveling wave solution for generalized Fisher equation and Lotka–Volterra (L-V) model with diffusion using hyperbolic function method. The Painleve’ analysis has been used to check both of the system’s integrability. Obtained solutions have also been plotted to represent their spatio-temporal dependence. The three dimensional plot shows a monotonic profile of the solutions.