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In this paper, we construct two families of new explicit exact traveling wave solutions of the Camassa–Holm equation which solve an open problem in the previous paper [Zhang G, Qiao ZJ, Liu F. Cuspons and smooth solitons of the Camassa-Holm equation under an inhomogeneous boundary condition. Pac J Appl Math. 2007;1(1):113–130]. With the help of the numerical graphs and a profound mathematical analysis, we discovered the analytic formulas for those two explicit exact solutions. [ABSTRACT FROM AUTHOR]

The generalized (G'/G)-expansion method with the aid of Maple is proposed to seek exact solutions of nonlinear evolution equations. For finding exact solutions are expressed three types of solutions that include hyperbolic function solution, trigonometric function solution, and rational solution. The article studies the Zakharov-Kuznetsov (ZK) equation, the generalized ZK (gZK) equation, and the generalized forms of these equations. Exact solutions with traveling wave solutions of nonlinear evolution equations are obtained. It is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

In this article, the exact solutions for nonlinear Drinfeld-Sokolov (DS) and generalized Drinfeld-Sokolov (gDS) equations are established. The rational Exp-function method (EFM) is used to construct solitary and soliton solutions of nonlinear evolution equations. This method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. Also, exact solutions with solitons and periodic structures are obtained. The obtained results are not only presented numerically but are also accompanied by insightful physical interpretations, enhancing the understanding of the complex dynamics described by these mathematical models. The utilization of the rational EFM and the broad spectrum of obtained solutions contribute to the depth and significance of this research in the field of nonlinear wave equations. [ABSTRACT FROM AUTHOR]

The objective of this present article is to study the two-component Novikov equation. This equation is also known as Geng–Xue equation and is characterized by cubic nonlinearities. By applying the multiplier method, new conserved quantities are found, and some traveling wave solutions are derived based on the (G ′ G) -expansion method. Furthermore, a numerical solution is determined by employing the Haar wavelet collocation (HWC) method. In order to confirm the accuracy, this numerical solution is compared with the solution obtained by (G ′ G) -expansion method. Additionally, the convergence analysis of the HWC method is also presented. [ABSTRACT FROM AUTHOR]

Parabolic Finslerian Allen–Cahn equation is a quasilinear parabolic equation on Finsler metric measure spaces. In this paper, we establish a Harnack inequality for solutions to this equation on compact Finsler metric measure spaces using geometric methods. Moreover, we give a degree 4 polynomial gradient estimate of the traveling wave solution on flat-curved spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of traveling wave solution for temporally discrete Lotka Volterra competitive system with delays. By using the cross iteration method and Schauder’s fixed point theorem, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The obtained results makes up and improves the results of the existence of traveling wave solutions for this systems. [ABSTRACT FROM AUTHOR]

This article studies the phase portraits, chaotic patterns, and traveling wave solutions of the fractional order generalized Pochhammer–Chree equation. First, the fractional order generalized Pochhammer–Chree equation is transformed into an ordinary differential equation. Second, the dynamic behavior is analyzed using the planar dynamical system, and some three-dimensional and two-dimensional phase portraits are drawn using Maple software to reflect its chaotic behaviors. Finally, many solutions were constructed using the polynomial complete discriminant system method, including rational, trigonometric, hyperbolic, Jacobian elliptic function, and implicit function solutions. Two-dimensional graphics, three-dimensional graphics, and contour plots of some solutions are drawn.

This study focuses on the propagation behavior of traveling wave solution in microcrystalline materials using the polynomial complete discriminant system method. By establishing a complete discriminant system, we systematically analyze the formation and evolution process of traveling wave solution in microcrystalline materials. Specifically, we apply the cubic polynomial extension to the strain wave equation to obtain more accurate analytical solutions. Additionally, two-dimensional, three-dimensional, and contour plots are generated to visually illustrate the characteristics of the obtained solutions, facilitating a more intuitive understanding of their physical significance. These findings not only help reveal the propagation mechanism of traveling wave solution but also provide a theoretical foundation for the application of microcrystalline materials.

The higher dimensional Fokas equation is the integrable expansion of the Davey–Stewartson and Kadomtsev–Petviashvili equations. In wave theory, the Fokas model plays a crucial role in explaining the physical phenomena of waves both inside and outside of water. The (4+1)-dimensional fractional-order Fokas equation is the subject of this article. Two effective approaches are employed to obtain the solutions for the considered equation: the generalized auxiliary equation technique and the G′/(bG′+G+a) technique. Several novel soliton solutions are obtained, including periodic solitary waves, bright solitons, and dark solitons. Various parametric values are employed to produce these new soliton waves at certain fractional order levels α. Furthermore, the bilinear version of the equation helps to develop its two-wave, three-wave, and multi-wave, as well as lump and rogue wave solutions. The properties of the solutions to the underlying problem are most effectively analyzed through the use of graphical representations. These outcomes and techniques can be used to study various fractional-order problems that emerge in wave theory, such as those in physics, hydraulics, optical technology, quantum mechanics, and plasma particles.

This paper focuses on the existence of periodic and solitary wave solutions in a generalized KdV equation with an arbitrarily high-order convection term which introduces a time delay in the nonlinearity. For the equation with two different local generic delay kernels, by applying geometric singular perturbation theory and analyzing the perturbation of a hyper-elliptic Hamiltonian system of arbitrary higher degree, we respectively prove the existence of one or two periodic wave solutions with certain wave speed in an open interval, depending on the degree. The existence of solitary wave solutions with certain wave speeds is also established by Melnikov’s method. Our results demonstrate that distributed delays and the degree of nonlinear term can influence the existence and number of traveling wave solutions with particular wave speeds. [ABSTRACT FROM AUTHOR]

The traveling wave solution of the generalized KP-MEW-Burgers equation is analyzed qualitatively in this paper. The equilibrium properties, including hypercritical pitchfork bifurcation and transcritical bifurcation of the planar system, are analyzed in detail in the full parameter space. The global structures of the traveling wave equation are described completely. Results show that, under certain parameters, the equivalent planar system has heteroclinic orbits, homoclinic orbits, and periodic orbits. The generalized KP-MEW-Burgers equation has solitary waves, periodic waves, kink waves, and anti-kink waves. Based on the Kosambi-Cartan-Chern (KCC) theory, Jacobi stability of the discussed equation also is explored. Results show Jacobi stability and Lyapunov stability of traveling wave solutions are not entirely consistent. Last but not least, the six dimensional nonlinear system transformed from the planar system with periodic disturbance is also discussed, including periodic, quasi-periodic, and chaotic behaviors. [ABSTRACT FROM AUTHOR]

In this paper, by using the bifurcation theory for dynamical system, we construct traveling wave solutions of a high-order nonlinear Schrödinger equation with a quintic nonlinearity. Firstly, based on wave variables, the equation is transformed into an ordinary differential equation. Then, under the parameter conditions, we obtain the Hamiltonian system and phase portraits. Finally, traveling wave solutions which contains solitary, periodic and kink wave solutions are constructed by integrating along the homoclinic or heteroclinic orbits. In addition, by choosing appropriate values to parameters, different types of structures of solutions can be displayed graphically. Moreover, the computational work and it's figures show that this technique is influential and efficient.

In this article, the phase portraits, chaotic patterns, and traveling wave solutions of the conformable extended Korteweg–de-Vries (KdV) model are investigated. First, the conformal fractional order extended KdV model is transformed into ordinary differential equation through traveling wave transformation. Second, two-dimensional (2D) planar dynamical system is presented and its chaotic behavior is studied by using the planar dynamical system method. Moreover, some three-dimensional (3D), 2D phase portraits and the Lyapunov exponent diagram are drawn. Finally, many meaningful solutions are constructed by using the complete discriminant system method, which include rational, trigonometric, hyperbolic, and Jacobi elliptic function solutions. In order to facilitate readers to see the impact of fractional order changes more intuitively, Maple software is used to draw 2D graphics, 3D graphics, density plots, contour plots, and comparison charts of some obtained solutions.

This article investigates the construction of new traveling wave solutions for the conformable fractional Klein–Gordon equation, which is a well‐known mathematical and physical model that can be used to explain spinless pion and de Broglie waves. In order to accomplish this task, a classic and effective analysis method, namely, the extended tanh–coth method, was utilized. By introducing appropriate transformations, the conformable fractional Klein–Gordon equations are reduced to ordinary differential equations, and then the solutions with hyperbolic function form are obtained. In addition, the effect of fractional parameters on waveform is analyzed by drawing two‐ and three‐dimensional graphics. These results contribute to a deeper understanding of the dynamics of the conformable fractional Klein–Gordon equation. The research in this article also indicates that the extended tanh–coth method is a straightforward and concise technique that has the potential to be applicable to many other conformable fractional partial differential equations that appear in mathematical physics. [ABSTRACT FROM AUTHOR]

This paper aims to study the existence of traveling wave solutions (TWS) for a three-component noncooperative systems with nonlocal diffusion. Our main results reveal that when a threshold ℜ > 1 , there exists a critical wave speed c ∗ > 0. By using sub- and super-solution methods and Schauder's fixed point theorem, we prove that the system admits a nontrivial TWS for each c ≥ c ∗ . Meanwhile, we show that there exists no nontrivial TWS for c < c ∗ by detailed analysis. Finally, we apply our results to a nonlocal diffusive epidemic model with vaccination, and the boundary asymptotic behavior of TWS for the special case is obtained by constructing a suitable Lyapunov functional. Our research provides some insights on how to deal with the problem of TWS for the nonlocal diffusive epidemic models with bilinear incidence, which extends some results in the previous studies. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discover analytically the interactional responses of populations in a dynamic region where the reaction–diffusion process with forcing effects takes place through traveling wave solutions. An expansion method is considered here to properly capture the responses for the first time. In order to profoundly analyze the physical and mathematical discussions, some illustrative behavioral results are exhibited for various values of physical parameters. Especially for the different values of diffusion coefficients in the model under consideration, their effects on the behavior of the solitary wave are discussed and observationally supported by considering various illustrations. It is also seen that the solutions representing the diffusion seen to be in the form of the behavior of hexagonal Turing patterns in different time periods. The application of this study in mathematical biology is to analyze the relationship between the population density of certain species in any local region and the specific population density with invasion characteristics. In addition, the formation of the extinction vortex of the invading population, depending on the characteristics of the solutions presented, is also descriptively discussed. [ABSTRACT FROM AUTHOR]

In this article, the phase portraits, chaotic patterns, and traveling wave solutions of the stochastic Sasa–Satsuma equation are investigated. Firstly, the stochastic Sasa–Satsuma equation is transformed into an ordinary differential equation through traveling wave transformation. Secondly, two-dimensional planar dynamical system is presented by using the theory of planar dynamical systems. Then, the three-dimensional and two-dimensional phase portraits of the dynamical system are drawn by using Maple software. Finally, the complete discriminant system method is used to solve the stochastic Sasa-Satsuma equation, resulting in many solutions that other methods cannot obtain, including rational, trigonometric, hyperbolic, and Jacobi elliptic function solutions. Moreover, three-dimensional-surface plots and two-dimensional-shape plots for the module length of some solutions under different parameters are drawn by using Maple software. The innovation of this article lies in introducing stochastic parameters into the Sasa–Satsuma equation, obtaining more diverse and comprehensive conclusions. [ABSTRACT FROM AUTHOR]

In this article, the dynamic behavior and solitary wave solutions of the Akbota equation were studied based on the analysis method of planar dynamic system. This method can not only analyze the dynamic behavior of a given equation, but also construct its solitary wave solution. Through traveling wave transformation, the Akbota equation can easily be transformed into an ordinary differential equation, and then into a two-dimensional dynamical system. By analyzing the two-dimensional dynamic system and its periodic disturbance system, planar phase portraits, three-dimensional phase portraits, Poincar'e sections, and sensitivity analysis diagrams were drawn. Additionally, Lyapunov exponent portrait of a dynamical system with periodic disturbances was drawn using mathematical software. According to the maximum Lyapunov exponent portrait, it can be deduced whether the system is chaotic or stable. Solitary wave solutions of the Akbota equation are presented. Moreover, a visualization diagram and contour graphs of the solitary wave solutions are presented. [ABSTRACT FROM AUTHOR]

The Schrödinger–Hirota equation is one of the most important models of contemporary physics which is popular throughout the broad fields of fluid movement as well as in the study of thick-water crests, liquid science, refractive optical components and so on. In this paper, we utilize the Hirota bilinear technique and the unified technique to attain various soliton solutions of the governing model analytically. These approaches are robust, powerful and unique also have many applications in different fields of mathematical physics. The solutions attained from these techniques are highly valuable and useful in various fields of sciences specially in the transmissions of optical fibers, also they give different behaviors including V-shaped and periodic soliton solution behavior. Further, the approaches applied here are not applied in this model previously. Therefore, ours is a new work, which summarizes its novelty. The 3D, 2D and contour plots are included to grasp the understanding of solutions' behavior. These findings are valuable in electronic communications such as elliptical circuits and in investigation of solitude controlling. [ABSTRACT FROM AUTHOR]

Motivated by the incompressible limit of a cell density model, we propose a free boundary tumor growth model where the pressure satisfies an obstacle problem on an evolving domain Ω (t) , and the coincidence set Λ (t) captures the emerging necrotic core. We contribute to the analytical characterization of the solution structure in the following two aspects. By deriving a semi-analytical solution and studying its dynamical behavior, we obtain quantitative transitional properties of the solution separating phases in the development of necrotic cores and establish its long time limit with the traveling wave solutions. Also, we prove the existence of traveling wave solutions incorporating non-zero outer densities outside the tumor bulk, provided that the size of the outer density is below a threshold. [ABSTRACT FROM AUTHOR]

A recent paper (Zhang R, Yu X. Traveling waves for a four‐compartment lattice epidemic system with exposed class and standard incidence. Math Meth Appl Sci. 2022; 45: 113‐136) presented a four‐compartment lattice epidemic system with exposed class and standard incidence. The authors studied the existence and non‐existence of traveling wave solutions of this system. However, the limit behavior of R$$ R $$‐component of traveling wave solutions is still open. In this paper, we solve this open problem and establish the boundedness of traveling wave solutions by analysis technique. Meanwhile, we study the non‐existence of traveling wave solutions with non‐positive wave velocity. [ABSTRACT FROM AUTHOR]

The main object of this paper is to study the traveling wave solutions of the fractional coupled Konopelchenko–Dubrovsky model by using the complete discriminant system method of polynomials. Firstly, the fractional coupled Konopelchenko–Dubrovsky model is simplified into nonlinear ordinary differential equations by using the traveling wave transformation. Secondly, the trigonometric function solutions, rational function solutions, solitary wave solutions and the elliptic function solutions of the fractional coupled Konopelchenko–Dubrovsky model are derived by means of the polynomial complete discriminant system method. Moreover, a two-dimensional phase portrait is drawn. Finally, a 3D-diagram and a 2D-diagram of the fractional coupled Konopelchenko–Dubrovsky model are plotted in Maple 2022 software. [ABSTRACT FROM AUTHOR]

This research paper introduces a novel approach to deriving traveling wave solutions (TWSs) for the Caputo-fractional Klein–Gordon equations. This research presents a distinct methodological advancement by introducing TWSs of a particular time-fractional partial differential equation, utilizing a non-local fractional operator, specifically the Caputo derivative. To achieve our goal, a novel transformation is considered, that converts a time-fractional partial differential equation into fractional ordinary differential equations, enabling analytical solutions through various analytical methods. This paper employs the homotopy analysis method to achieve the target objectives. To demonstrate the efficiency and applicability of the proposed transform and method, two examples are discussed and analyzed in figures. [ABSTRACT FROM AUTHOR]

In this paper, within the scope of the local fractional derivative theory, the (4+1)-dimensional local fractional Fokas equation is researched. The study of exact solutions of high-dimensional nonlinear partial differential equations plays an important role in understanding complex physical phenomena in reality. In this paper, the exact traveling wave solution of generalized functions is analyzed defined on Cantor sets in high-dimensional integrable systems. The results of non-differentiable solutions in different cases are numerically simulated when the fractal dimension is equal to ρ = ln 2/ln 3. The results show that the exact solution of the local fractional Fokas equation represents the fractal waves on the shallow water surface. Through numerical simulation, we find that the exact solution of the local fractional Fokas equation can describe the fractal waves and waves characteristics of shallow water surface. It also shows that the study of traveling wave solutions of nonlinear local fractional equations has important significance in mathematical physics. [ABSTRACT FROM AUTHOR]

This paper investigates the pure-cubic complex Ginzburg–Landau equation (PC-CGLE) with different nonlinearities such as Kerr law, power law and so on. We get the dynamic systems and show that solitons and periodic solutions exist through the complete discrimination system for the polynomial method (CDSPM). To verify these conclusions, we construct the traveling wave solution via the CDSPM, and some new solutions are also built. The soliton stability and modulation instability with two types of nonlinearities are discussed. Finally, by adding perturbed terms to the dynamic system, we obtain the largest Lyapunov exponents and the phase diagrams of the equation, which proves there are chaotic behaviors in PC-CGLE. The results such as Gaussian soliton solutions and chaotic behavior for PC-CGLE are initially discovered in the present paper.

In this paper, we considered a delayed predator-prey model with stage structure and Beddington-DeAngelis type functional response. First, we analyzed the stability of the constant equilibrium points of the model by the linear stability method. Furthermore, we considered the existence of traveling wave solutions connecting the zero equilibrium point and the unique positive equilibrium point. Second, we transformed the existence of traveling wave solutions into the existence of fixed points of an operator by constructing suitable upper and lower solutions, and combined with the Schauder fixed point theorem, we gave the existence of fixed points and obtained the existence of traveling wave solutions of the model.

We explored the (3+1)-dimensional negative-order Korteweg-de Vries-alogero-Bogoyavlenskii-Schiff (KdV-CBS) equation, which develops the classical Korteweg-de Vries (KdV) equation and extends the contents of nonlinear partial differential equations. A traveling wave transformation is employed to transform the partial differential equation into a system of ordinary differential equations linked with a cubic polynomial. Utilizing the complete discriminant system for polynomial method, the roots of the cubic polynomial were classified. Through this approach, a series of exact solutions for the KdV-CBS equation were derived, encompassing rational function solutions, Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. These solutions not only simplified and expedited the process of solving the equation but also provide concrete and insightful expressions for phenomena such as optical solitons. Presenting these obtained solutions through 3D, 2D, and contour plots offers researchers a deeper understanding of the properties of the model and allows them to better grasp the physical characteristics associated with the studied model. This research not only provides a new perspective for the in-depth exploration of theoretical aspects but also offers valuable guidance for the practical application and advancement of related technologies.

In general, Schrödinger's equations have lots of applications in quantum mechanics. Accordingly, in this paper, some new forms of the exact traveling wave solutions have been obtained for two different types of Schrödinger's equations, namely the unstable generalized nonlinear Schrödinger's equation and the hyperbolic nonlinear Schrödinger's equation. In the literature one can see different forms of the solution by various existing methods. However, the applied methodology or obtained results are sometimes complicated in nature. So here to obtain the new and simple forms of the solutions for both of the equations we have used a very simple and important method known as sine-cosine method. In addition, the probability density functions (PDFs) for both of the considered problems have also been computed. To visualize the impact of the solutions, graphical representations have been made with respect to various parameters. From the results one can see, they are in terms of complex-valued functions. In special cases comparison has also been given. Furthermore, the results are computed and validated using the Maple software. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analytical study of traveling wave solutions to a generalized Boussinesq equation with nonlinear dispersion. Utilizing the bifurcation method of dynamical systems, the existence of compacton and peakon solutions are established. Under certain parameter conditions, the compacton and peakon solutions can be bifurcated by smooth periodic wave solutions, smooth solitary wave solutions, and singular cusp solutions. Numerical simulations are supplied to corroborate the analytical results. Some previous results are extended. [ABSTRACT FROM AUTHOR]

In this paper, we considered a delayed predator-prey model with stage structure and Beddington-DeAngelis type functional response. First, we analyzed the stability of the constant equilibrium points of the model by the linear stability method. Furthermore, we considered the existence of traveling wave solutions connecting the zero equilibrium point and the unique positive equilibrium point. Second, we transformed the existence of traveling wave solutions into the existence of fixed points of an operator by constructing suitable upper and lower solutions, and combined with the Schauder fixed point theorem, we gave the existence of fixed points and obtained the existence of traveling wave solutions of the model. [ABSTRACT FROM AUTHOR]

Study on solitary wave phenomenon are closely related on the dynamics of the plasma and optical fiber system, which carry on broad range of wave propagation. The space–time fractional modified Benjamin–Bona–Mahony equation and Duffing model are important modeling equations in acoustic gravity waves, cold plasma waves, quantum plasma in mechanics, elastic media in nonlinear optics, and the damping of material waves. This study has effectively developed analytical wave solutions to the aforementioned models, which may have significant consequences for characterizing the nonlinear dynamical behavior related to the phenomenon. Conformable derivatives are used to narrate the fractional derivatives. The expanded tanh-function method is used to look into such kinds of resolutions. An ansatz for analytical traveling wave solutions of certain nonlinear evolution equations was originally a power sequence in tanh. The discovered explanations are useful, reliable, and applicable to chaotic vibrations, problems of optimal control, bifurcations to global and local, also resonances, as well as fusion and fission phenomena in solitons, scalar electrodynamics, the relation of relativistic energy–momentum, electromagnetic interactions, theory of one-particle quantum relativistic, and cold plasm. The solutions are drafted in 3D, contour, listpoint, and 2D patterns, and include multiple solitons, bell shape, kink type, single soliton, compaction solitary wave, and additional sorts of solutions. With the aid of Maple and MATHEMATICA, these solutions were verified and discovered that they were correct. The mentioned method applied for solving NLFPDEs has been designed to be practical, straightforward, rapid, and easy to use. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a variety of analytical wave solutions of a (2 + 1)-dimensional extended shallow water wave equation. The applications of the governing equation are enormous in ocean modeling and investigation of moist-convection properties in atmospheric dynamics. Two powerful mathematical approaches, the exp (− Υ (g)) -expansion method (EEM) and the generalized projective Riccati equations method (GPREM), were used to find closed-form traveling wave solutions. In addition, the breather wave solutions were described using the Cole-Hopf transform, derived with the help of Hirota bilinear method (HBM). Eventually, we obtain 46 closed-form solutions in explicit form. The general form of obtained solutions include trigonometric, rational, exponential and hyperbolic function solutions. To illustrate the physical significance of some novel results, we provided contour, two- and three-dimensional graphics under the suitable free choices of unknown parameters. The dynamical shapes of these solutions are one soliton, multiple solitons, kink, anti-kink, lump and periodic solutions. We believe that the applied methods of this work are well-organized, genuine and powerful mathematical tools for solving the nonlinear evolution equations (NLEEs) occurring in the study of ocean science and engineering. [ABSTRACT FROM AUTHOR]

To provide insights into the spreading speed and propagation dynamics of viruses within a host, in this paper, we investigate the traveling wave solutions and minimal wave speed for a degenerate viral infection dynamical model with a nonlocal dispersal operator and saturated incidence rate. It is found that the minimal wave speed c ∗ is the threshold that determines the existence of traveling wave solutions. The existence of traveling fronts connecting a virus-free steady state and a positive steady state with wave speed c ≥ c ∗ is established by using Schauder's fixed-point theorem, limiting arguments, and the Lyapunov functional. The nonexistence of traveling fronts for c < c ∗ is proven by the Laplace transform. In particular, the lower-bound estimation of the traveling wave solutions is provided by adopting a rescaling method and the comparison principle, which is a crucial prerequisite for demonstrating that the traveling semifronts connect to the positive steady state at positive infinity by using the Lyapunov method and is a challenge for some nonlocal models. Moreover, simulations show that the asymptotic spreading speed may be larger than the minimal wave speed and the spread of the virus may be postponed if the diffusion ability or diffusion radius decreases. The spreading speed may be underestimated or overestimated if local dispersal is adopted. [ABSTRACT FROM AUTHOR]

Phages are ubiquitous in nature, but many essential factors of host-phage biology have not yet been integrated into mathematical models. In this paper, we investigate a spatial phage-bacteria model to describe the propagation of phages and bacteria in different types of nutrient media. Unlike existing models, we construct a more realistic reaction-diffusion model that incorporates inoculum and bacterial growth and movement, then rigorous mathematical analysis is challenging. We study traveling wave solutions and obtain complete information about the existence and nonexistence of nontrivial traveling wave solutions. The threshold conditions for the existence and nonexistence of traveling wave solutions are obtained by using Schauder's fixed point theorem, limiting argument, and one-sided Laplace transform. Considering different propagation media, we extend the existence of traveling wave solutions from liquid nutrition model to agar model. Moreover, in the absence of bacterial mortality, we obtain the existence of a new traveling wave solution describing phage invasion. We attempt to explain the occurrence of co-transport by the existence and nonexistence of traveling waves, and screen out the key parameters affecting the co-transport of phages and bacteria according to the definition of critical wave speed. Finally, we provide numerical simulations to verify the theoretical results and reveal the effects of key parameters on the propagation of phages and bacteria. [ABSTRACT FROM AUTHOR]

In this paper, we acquire novel traveling wave solutions of the generalized seventh-order Korteweg-de Vries equation and the seventh-order Kawahara equation as a special case with physical interest. Primarily, we use the advanced exp(φ,ξ)-expansion method to find new exact solutions of the first equation, by considering two auxiliary equations. Then, we attain some exact solutions of the seventh-order Kawahara equation by using this method with another auxiliary equation, and also using the modified (G'/G)-expansion method, where G satisfies a second-order linear ordinary differential equation. Additionally, utilizing the recent scientific instruments, the 2D, 3D, and contour plots are displayed. The solutions obtained in this paper include bright solitons, dark solitary wave solutions, and multiple dark solitary wave solutions. It is shown that these two methods provide an effective mathematical tool for solving nonlinear evolution equations arising in mathematical physics and engineering. [ABSTRACT FROM AUTHOR]

This paper is devoted to a three-species stochastic competitive system with multiplicative noise. The existence of stochastic traveling wave solution can be obtained by constructing sup/sub-solution and using random dynamical system theory. Furthermore, under a more restrict assumption on the coefficients and by applying Feynman-Kac formula, the upper/lower bounds of asymptotic wave speed can be achieved. [ABSTRACT FROM AUTHOR]

We explored the (3+1)-dimensional negative-order Korteweg-de Vries-alogero-Bogoyavlenskii-Schiff (KdV-CBS) equation, which develops the classical Korteweg-de Vries (KdV) equation and extends the contents of nonlinear partial differential equations. A traveling wave transformation is employed to transform the partial differential equation into a system of ordinary differential equations linked with a cubic polynomial. Utilizing the complete discriminant system for polynomial method, the roots of the cubic polynomial were classified. Through this approach, a series of exact solutions for the KdV-CBS equation were derived, encompassing rational function solutions, Jacobi elliptic function solutions, hyperbolic function solutions, and trigonometric function solutions. These solutions not only simplified and expedited the process of solving the equation but also provide concrete and insightful expressions for phenomena such as optical solitons. Presenting these obtained solutions through 3D, 2D, and contour plots offers researchers a deeper understanding of the properties of the model and allows them to better grasp the physical characteristics associated with the studied model. This research not only provides a new perspective for the in-depth exploration of theoretical aspects but also offers valuable guidance for the practical application and advancement of related technologies. [ABSTRACT FROM AUTHOR]

We derive a reaction–diffusion model with time-delayed nonlocal effects to study an epidemic's spatial spread numerically. The model describes infected individuals in the latent period using a structured model with diffusion. The epidemic model assumes that infectious individuals are subject to containment measures. To simulate the model in two-dimensional space, we use the continuous Runge–Kutta method of the fourth order and the discrete Runge–Kutta method of the third order with six stages. The numerical results admit the existence of traveling wave solutions for the proposed model. We use the COVID-19 epidemic to conduct numerical experiments and investigate the minimal speed of spread of the traveling wave front. The minimal spreading speeds of COVID-19 are found and discussed. Also, we assess the power of containment measures to contain the epidemic. The results depict a clear drop in the spreading speed of the traveling wave front after applying containment measures to at-risk populations. [ABSTRACT FROM AUTHOR]

The behaviors of wave solutions of the fractional nonlinear space-time Sharma-Tasso-Olever equation and the fractional nonlinear space-time Estevez-Mansfield-Clarkson equation, representing a fluid dynamics equation and a shallow water equation, respectively, can be obtained by transforming the fractional nonlinear space-time partial differential equations into nonlinear ordinary differential equations with the Jumarie's Riemann-Liouville fractional derivative, and solving for a finite series form of solution in the Riccati sub-equation method. The newly discovered traveling wave solutions took the forms of generalized triangular functions and generalized hyperbolic functions, which ultimately led to the assessing physical wave behaviors. These behaviors are manifested in kink and periodic waves, and they are separately depicted by 2-D, 3-D and contour graphs. In addition, the results we received were more diverse than previous solutions.

By using the Riccati-Bernoulli (RB) subsidiary ordinary differential equation method, we proposed to solve kink-type envelope solitary solutions, periodical wave solutions and exact traveling wave solutions for the coupled Higgs field (CHF) equation. We get many solutions by applying the Bäcklund transformations of the CHF equation. The proposed method is simple and efficient. In fact, we can deal with some other classes of nonlinear partial differential equations (NLPDEs) in this manner.

Through the application of the deformation algorithm, a novel (3+1)-dimensional Gardner equation and its associated Lax pair are derived from the (1+1)-dimensional Gardner equation and its conservation laws. As soon as the (3+1)-dimensional Gardner equation is set to be y or z independent, the Gardner equations in (2+1)-dimension are also obtained. To seek the exact solutions for these higher dimensional equations, the traveling wave method and the symmetry theory are introduced. Hence, the implicit expressions of traveling wave solutions to the (3+1)-dimensional and (2+1)-dimensional Gardner equations, the Lie point symmetry and the group invariant solutions to the (3+1)-dimensional Gardner equation are well investigated. In particular, after selecting some specific parameters, both the traveling wave solutions and the symmetry reduction solutions of hyperbolic function form are given. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigate the existence and nonexistence of traveling wave solution for a diffusive vector-borne disease model with nonlocal reaction and distributed delays. We demonstrate that the basic reproduction number R 0 of the corresponding ordinary differential equation system as a threshold determines whether the model admits traveling waves or not and there exists a critical wave speed c m ∗ > 0 when R 0 > 1 . Specifically, (i) As R 0 > 1 and the wave speed c > c m ∗ , the existence of traveling waves for the system is established with the aid of a perturbed system; (ii) As R 0 > 1 and 0 < c < c m ∗ , the nonexistence of traveling waves is proved via the two-sided Laplace transform; (iii) As R 0 ≤ 1 and c > 0 , the nonexistence is obtained by utilizing the comparison principle. The theoretical results are applied to dengue fever epidemics. We study the effects of geographical movement, nonlocal interaction, incubation period and R 0 on the threshold speed c m ∗ for dengue fever. [ABSTRACT FROM AUTHOR]

Bifurcation of non-monotone traveling wave solutions of the three-species Lotka-Volterra competition diffusion system under strong competition is studied. The well-known front and back traveling wave formed by two species may lose its stability by the effect of third species and, as a result, allows the invasion. To discuss how the invasion is possible, stability change with respect to the intrinsic growth rate for the alien species are studied. Both numerical and theoretical bifurcation analysis around the bifurcation point reveal how the invasion affects the segregation of the original two species. [ABSTRACT FROM AUTHOR]

The behaviors of wave solutions of the fractional nonlinear space-time Sharma-Tasso-Olever equation and the fractional nonlinear space-time Estevez-Mansfield-Clarkson equation, representing a fluid dynamics equation and a shallow water equation, respectively, can be obtained by transforming the fractional nonlinear space-time partial differential equations into nonlinear ordinary differential equations with the Jumarie's Riemann-Liouville fractional derivative, and solving for a finite series form of solution in the Riccati sub-equation method. The newly discovered traveling wave solutions took the forms of generalized triangular functions and generalized hyperbolic functions, which ultimately led to the assessing physical wave behaviors. These behaviors are manifested in kink and periodic waves, and they are separately depicted by 2-D, 3-D and contour graphs. In addition, the results we received were more diverse than previous solutions. [ABSTRACT FROM AUTHOR]