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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]

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]

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]

This paper is concerned with the propagation dynamics of a discrete diffusive equation with non‐local delay. First of all, by exploring the asymptotic behavior of the solution of the upper system corresponding to the perturbation equation, we obtain the global stability of semi‐wavefronts. It is noteworthy that the kernel function β(k)$$ \beta (k) $$ can be asymmetric (i.e., β(k)≠β(−k)$$ \beta (k)\ne \beta \left(-k\right) $$). Secondly, we estimate the level set of the equation on the basis of the above stability. In particular, the results show that the solution of the compact supported initial value problem is not persistent in special case. [ABSTRACT FROM AUTHOR]

The (2+1)$$ \left(2+1\right) $$‐dimensional coupled cubic–quintic complex Ginzburg–Landau equations ( (2+1)$$ \left(2+1\right) $$‐DCC‐QCGLEs) can simulate a variety of binary fluid thermal convection characteristics, containing complex parameters. The analysis of pattern formation in chaotic and nonlinear dynamical systems can benefit greatly from the use of this thermal convection model. The primary goal of this study is to find analytical solutions for some recent advances that have been made for Rayleigh–Bénard convection by applying the (2+1)$$ \left(2+1\right) $$‐DCC‐QCGLEs model for slowly varying spatio‐temporal amplitudes of the wave motion. In addition, novel traveling solitary wave solutions for the model equation are derived using a very useful method to investigate how complex physical coefficients affect the profiles of propagating waves. Furthermore, we introduce the WTC‐Kruskal algorithm of the Painlevé methodology to examine the integrability of the (2+1)$$ \left(2+1\right) $$‐DCC‐QCGLEs and the truncated Painlevé expansion is used to extract the Bäcklund transform, from which new solitary solutions can be acquired. The results also demonstrated a good agreement with previous works and were more significant and accurate in two and three dimensions of the proposed model. Finally, the computational results indicate that the effects of the physical parameters of the considered equations can be demonstrated by utilizing 2D and 3D graphics for different values of these parameters. [ABSTRACT FROM AUTHOR]

Infectious illnesses have an exceptional affect on the economic system and society. Dynamic models of infectious illnesses are high quality devices for revealing the legal guidelines of illnesses transmission. Quarantine and nonlinear innate immunity are the crucial elements in the manage of infectious illnesses. However, there are no mathematical models that comprehensively find out about the impact of each innate immunity and quarantine. In this paper, we investigate a system of nonlocal partial differential equations related to an epidemic model. Under appropriate hypothesis on the relaxation function and the kernel of the equation, we prove that the traveling wave solutions are globally exponentially stable by applying the variable exponent theory combining with adequate variational methods and a variant of the mountain pass lemma. We also obtain the uniqueness of traveling wave solutions. [ABSTRACT FROM AUTHOR]

Nonlinear electrical lattices are powerful experimental tool for studying nonlinear dispersive media and creating opportunities for the realistic modeling of electrical solitons. In this work, we adopt the generalized auxiliary equation methods, the first integral method, and the Möbius transformation approach to extract a variety of traveling and solitary wave solutions of the nonlinear Salerno equation describing the nonlinear discrete electrical lattice with nonlinear dispersion. The kink, antikink, dark, bright, peakons, antipeakons, and periodic wave solutions are all derived. The existence criteria of solitons are described. It has been shown that for the selective values of arbitrary constants in an auxiliary equation, the generalized auxiliary equation method II is reduced to the new ϕ6$$ {\phi}^6 $$‐model expansion method as well as the new extended auxiliary equation method. The impact of free parameters on the obtained solutions is investigated and graphically depicted using physical descriptions. The acquired results are important for the validity of numerical and experimental results and further understanding of the wave propagation in the nonlinear discrete electrical lattice. In addition, the stability analysis of the obtained solitary wave solutions is studied. Furthermore, using the maple software, all derived solutions were checked by re‐entering them into the considered equation. [ABSTRACT FROM AUTHOR]

In this work, the local fractional Zakharov–Kuznetsov‐modified equal width dynamical (LFZKMEWD) model is investigated on Cantor sets by using the local fractional derivative (LFD). The fractal variational wave method (FVWM) is employed to obtain the exact traveling wave solutions of the nondifferentiable type for the LFZKMEW model. The numerical example illustrates the FVWM is efficient and straightforward. The properties of exact traveling wave solutions are also elaborated by some figures. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with traveling waves in a nonlocal delayed epidemic model with diffusion. Firstly, by considering a six‐dimensional nondelayed system with the help of variable transformation, we establish the existence of monotone traveling waves by means of the abstract theory, which implies the existence of traveling waves connecting the disease‐free equilibrium and the epidemic coexistence equilibrium. Secondly, we prove the global stability of traveling waves based on spectral analysis method, which reveals that the solutions of the initial values and the traveling waves are exponentially close. Thirdly, we obtain that the wave speed is unique by choosing suitable parameters to construct new upper and lower solutions, which shows that the bistable waves keep the uniqueness of the wave speeds in the case of nonlocal delays. As an application of our results, we give a special example. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and nonexistence of a nonlocal dispersal cholera model with vaccination. First, we explore the existence of traveling wave solution when R0 > 1 and c ≥ c∗ by using the Schauder's fixed‐point theorem associated with the upper‐lower solutions. Moreover, the Lyapunov functional is used to show the boundary asymptotic behavior of traveling wave solution. Furthermore, in the case when R0 > 1 and c < c∗, we show that the model system has nonexistence of traveling wave solution on the basis of the Laplace transform. At last, we discuss how the spatial movement and vaccination affect the minimal wave speed. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of traveling wave solutions in nonlocal delayed reaction‐diffusion systems with partial quasimonotonicity. The method is based on Schauder's fixed point theorem and the cross‐iteration scheme. We also present some applications of our results to diffusive and competition‐cooperation system with nonlocal delays by choosing 3 different kernels. [ABSTRACT FROM AUTHOR]

In this paper, combining with a new generalized ansätz and the fractional Jacobi elliptic equation, an improved fractional Jacobi elliptic equation method is proposed for seeking exact solutions of space-time fractional partial differential equations. The fractional derivative used here is the modified Riemann-Liouville derivative. For illustrating the validity of this method, we apply it to solve the space-time fractional Fokas equation and the the space-time fractional BBM equation. As a result, some new general exact solutions expressed in various forms including the solitary wave solutions, the periodic wave solutions, and Jacobi elliptic functions solutions for the two equations are found with the aid of mathematical software Maple. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of traveling wave solutions for n-dimensional delayed reaction-diffusion systems. By using Schauder's fixed point theorem, we establish the existence result of a traveling wave solution connecting two steady states by constructing a pair of upper-lower solutions that are easy to construct. As an application, we apply our main results to a four-dimensional delayed predator-prey system and obtain the existence of traveling wave solutions. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

When one characteristic of the system is linearly degenerate, under suitable boundary conditions, we get the existence of traveling wave solutions located on the corresponding characteristic trajectory to the one-sided mixed initial-boundary value problem. When the system is linearly degenerate, by introducing the semi-global normalized coordinates, we derive the related formulas of wave decomposition to prove the stability of traveling wave solutions corresponding to all leftward and the rightmost characteristic trajectories. Finally, for the traveling wave solutions corresponding to other rightward characteristic trajectories, some examples show their possible instability. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]