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By the combination of Lie symmetry analysis and dynamical system method, the (2+1)-dimensional dissipative long wave system is studied. First, we get Lie algebra and Lie symmetry group of the system. Then, by using the dynamical system method, the bifurcation and phase portraits of the corresponding traveling system of the system are obtained, it is shown that for different parametric space, the system has infinitely many solitary wave solutions, periodic wave solutions, kink or anti kink wave solutions. At last, the conservation laws of the system are given. [ABSTRACT FROM AUTHOR]

In the mathematical modeling of solid fuel combustion, thermal losses are commonly neglected to allow a rigorous analysis. In this work, different choice of simplifications was made, making the model more physically realistic. Besides the analysis of the solutions including existence and uniqueness, a classification in the combustion regimes as a function of thermal losses was obtained. Some examples were analyzed, evidencing the validity of the proposed classification. Numerical simulations with more realistic models show the validity of the proposed formulas and applications for filtration combustion. [ABSTRACT FROM AUTHOR]

In this paper, we use Schauder's fixed point theorem to establish the existence of invasion waves in a stage-structured competitive system on higher-dimensional lattices. To illustrate our results, we construct a pair of upper and lower solutions. [ABSTRACT FROM AUTHOR]

The main goal of this study is to obtain the traveling wave solutions for Regularized Long Wave (RLW) equation by using (1/G')-expansion method. By giving special values to the constants in the solutions obtained, 3D, 2D, and contour graphics are presented. These graphics are a special solution of the (RLW) equation, and they represent a stationary wave of the equation. A computer package program is used to find the solutions and graphics presented in this article. [ABSTRACT FROM AUTHOR]

In this paper, we study the entire solutions of the Fisher–KPP (Kolmogorov–Petrovsky–Piskunov) equation ut = uxx + f(u) on the half line [0, ∞) with Dirichlet boundary condition at x = 0. (1) For any $c \ge 2\sqrt {f'(0)} $ , we show the existence of an entire solution ${{\cal U}^c}(x,t)$ which connects the traveling wave solution φc(x + ct) at t = −∞ and the unique positive stationary solution V(x) at t = +∞; (2) We also construct an entire solution ${{\cal U}}(x,t)$ which connects the solution of ηt = f(η) at t = −∞ and V(x) at t = +∞. [ABSTRACT FROM AUTHOR]

In this article, we employ the nonlinear complex conformable Kundu–Eckhaus model which represents the propagation of the ultra-short femtosecond pulses and the rogue wave in an optical fiber. The exact and solitary wave solutions of this model are obtained for the first time in the framework of the modified extended tanh-function method. The obtained results give a good interpretation for the propagation of ultra-short femtosecond pulses and the rogue wave in an optical fiber. A comparison between our result and that obtained in previous work is listed. The constructed method can be applied to many other nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

This article concerns traveling wave solutions of the fully parabolic Keller-Segel chemotaxis system with logistic source, ... where χ, µ, λ, a, b are positive numbers, and τ ≥ 0. Among others, it is proved that if b > 2χµ and τ ≥ 1/2 (1 - λ/α)+, then for every c ≥ 2√α, this system has a traveling wave solution (u; v)(t; x) = (Uτ,c(x ⋅ξ - ct), Vτ,c (x ⋅ ξ - ct)) (for all ξ ∈ ℝN) connecting the two constant steady states (0; 0) and (a/b, µ/λ, a/b), and there is no such solutions with speed c less than 2√a, which improves the results established in [30], and shows that this system has a minimal wave speed c0* = 2√a, which is independent of the chemotaxis. [ABSTRACT FROM AUTHOR]

In this paper, we study the traveling wave solutions of a Lotka-Volterra diffusion competition system with nonlocal terms. We prove that there exists traveling wave solutions of the system connecting equilibrium (0,0) to some unknown positive steady state for wave speed and there is no such traveling wave solutions for , where and respectively corresponds to the diffusion coefficients and intrinsic rate of an competition species. Furthermore, we also demonstrate the unknown steady state just is the positive equilibrium of the system when the nonlocal delays only appears in the interspecific competition term, which implies that the nonlocal delay appearing in the interspecific competition terms does not affect the existence of traveling wave solutions. Finally, for a specific kernel function, some numerical simulations are given to show that the traveling wave solutions may connect the zero equilibrium to a periodic steady state. [ABSTRACT FROM AUTHOR]

In this paper, a reaction-diffusion model with a cyclic structure is studied, which includes the SIS disease-transmission model and the nutrient-phytoplankton model. The minimal wave speed c∗ of traveling wave solutions is given. The existence of traveling semi-fronts with c > c∗ is proved by Schauder's fixed-point theorem. The traveling semi-fronts are shown to be bounded by rescaling method and comparison principle. The existence of traveling semi-front with c = c∗ is obtained by limit arguments. Finally, the traveling semi-fronts are shown to connect to the positive equilibrium by a Lyapunov function. [ABSTRACT FROM AUTHOR]

This paper presents a unified method to investigate exact traveling wave solutions of the nonlinear fractional-order and integer-order partial differential equations. We use the conformable fractional derivatives. The method is based on the bifurcation theory of planar dynamical systems. To show the effectiveness of this method, we choose Biswas–Milovic (for short, BM) equation with conformable derivative as an application. Also comparison is presented for the exact traveling wave solutions between the integer-order BM equation and fractional-order BM equation. It is believed that this approach can be extended to other nonlinear fractional-order partial differential equations. [ABSTRACT FROM AUTHOR]

In this paper, an analytical method called an exp(-φ(Ν))-expansion method is applied to find new exact traveling wave solution of a class of the fifth-order Korteweg-de Vries equation (fKdV), and some particular cases of this class have an aspect interesting physical that are the Lax, Sawada- Kotera, Caudrey-Dodd-Gibbon, Kaup- Kupershmidt and Ito equations. [ABSTRACT FROM AUTHOR]

In this paper we study traveling wave solutions of a non-cooperative lattice-diffusion system with time delay, which includes predator–prey models and disease-transmission models. Minimal wave speed of traveling wave solutions is given. Schauder's fixed-point theorem is applied to show the existence of semi-traveling wave solutions. The boundness and persistence of traveling wave solutions are overcome by using rescaling method and Laplace transform, where the application of Laplace transform to persistence is very novel and creative. The traveling wave solutions for some specific models are shown to connect to a positive equilibrium by using Lyapunov function and LaSalle's invariance principle. [ABSTRACT FROM AUTHOR]

In this paper, the traveling wave solution of Kerner konhauser model is obtained by phase plane analysis method, and a new model suitable for stability analysis is obtained by variable transformation. The new model can describe and predict the nonlinear traffic phenomenon on Expressway from the perspective of system global stability. In this paper, the nonlinear system and linear system are obtained respectively by traveling wave substitution and Taylor expansion at the equilibrium point. According to the qualitative theory of differential equations, the equilibrium point type and stability of the linear system are determined. Finally, the simulation results show the consistency between the numerical results and the theoretical analysis.

In this paper, the phase plane analysis method is used, which can describe and predict the nonlinear traffic phenomenon on Expressway from the perspective of system global stability. The nonlinear system of the model is obtained by the traveling wave substitution and Taylor expansion of the model, and the equilibrium point of the model is solved by specifying the model parameters. According to the definite theory of differential equation, the model is further analyzed to judge the type and stability of equilibrium point. Finally, numerical verification is carried out according to the simulation diagram. Through numerical verification, the simulation results of the model are consistent with the conclusions of theoretical analysis. It can more clearly describe the change of density or speed with time or road section.

In this paper we consider the spreading phenomena in the Fisher-KPP equation in a high dimensional cone with Dirichlet boundary condition. We show that any solution starting from a nonnegative and compact supported initial data spreads and converges to the unique positive steady state. Moreover, the asymptotic spreading speeds of the front in all directions pointing to the opening are c 0 (which is the minimal speed of the traveling wave solutions of the 1-dimensional Fisher-KPP equation). Surprisingly, they do not depend on the shape of the cone, the propagating directions and the boundary condition. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of traveling wave solutions in delayed higher dimensional lattice differential systems with partial monotonicity. By relaxing the monotonicity of the upper solutions and allowing it greater than positive equilibrium point, we establish the existence of traveling wave solutions by means of Schauder's fixed point theorem. And then, we apply our results to delayed competition‐cooperation systems on higher dimensional lattices. [ABSTRACT FROM AUTHOR]

In this paper, we consider the model of a long waves–short waves, which model the resonant interaction between the ocean wave, to investigate the qualitative behavior of traveling wave solutions of dynamical systems. To understand the underlying complex dynamics, we apply the bifurcation method, that serve as a mechanism for fascinating the physical behavior such as solitons and we obtain all possible phase portraits of the system under various conditions depending on the parameters associated with the planar dynamical system. [ABSTRACT FROM AUTHOR]

This paper mainly focuses on the entire solutions of a nonlocal dispersal equation with asymmetric kernel and bistable nonlinearity. Compared with symmetric case, the asymmetry of the dispersal kernel function makes more diverse types of entire solutions since it can affect the sign of the wave speeds and the symmetry of the corresponding nonincreasing and nondecreasing traveling waves. We divide the bistable case into two monostable cases by restricting the range of the variable, and obtain some merging-front entire solutions which behave as the coupling of monostable and bistable waves. Before this, we characterize the classification of the wave speeds so that the entire solutions can be constructed more clearly. Especially, we investigate the influence of the asymmetry of the kernel on the minimal and maximal wave speeds. [ABSTRACT FROM AUTHOR]

In this paper, the extended rational sinh-cosh method (ERSCM) and modified Khater method are applied to the biological population model to derive new exact solutions. Moreover, the stability property of some obtained solutions is discussed to show the ability of them for using in the model's applications. Implementation of the direct algebraic methods, the equations derived by substitution of the predicted solution are solved. It is significant to point out that new traveling wave solutions are found. The present methods are easy to employ and sufficient to determine the solutions. [ABSTRACT FROM AUTHOR]

The traveling wave solutions connecting two equilibria for a delayed Logistic equation in a cylinder are obtained for any delay τ > 0. We attain our goal by using the approach based on the combination of Schauder fixed point theory and the weak coupled upper–lower solutions method. Moreover, we prove that there is a constant c ∗ that serves as the minimal wave speed of such traveling wave solutions. [ABSTRACT FROM AUTHOR]

We consider the mathematical model with arbitrary power of nonlinearity which is described by the generalized Schrödinger equation. The Cauchy problem for this equation is not solved by the inverse scattering transform and we use the traveling wave reduction for the nonlinear partial differential equation. Using the traveling wave solutions we find the first integrals for the system of equations corresponding to real and imaginary parts of the profile pulse for a complex function. The system of equations is reduced by means of transformations to the first-order nonlinear ordinary differential equations with solutions expressing via the Weierstrass and Jacobi elliptic functions. The influence of the nonlinearity degree on the structure of periodic and solitary waves is studied. It is demonstrated that the degree of nonlinearity allows to control the amplitude and pulse length of periodic and solitary waves. [ABSTRACT FROM AUTHOR]

We consider traveling wave solutions of the generalized cubic-quantic Schrödinger equation used for description of the propagation pulses in optical fibers. We show that the system of equations for real and imaginary parts of dependent variable has two first integrals. These first integrals allow us to transform the system of equations to the nonlinear first-order ordinary differential equation of second power. We could not obtain the general solution of equation in general case. However we get the general solutions of equation expressing via the Weierstrass function at some additional conditions on the parameters of mathematical model. [ABSTRACT FROM AUTHOR]

The existence of traveling wave solutions for nonlocal dispersal SIR epidemic models with delay was studied. Firstly, the boundedness of I was proved by contradiction. Then according to the boundedness of I, the existence of traveling waves with c>c* was established. Secondly, through further analysis of traveling waves with super-critical speeds, the existence of traveling waves with the critical speed was derived. Finally, the influence of basic reproduction number R0 on the existence of c>c* was discussed. [ABSTRACT FROM AUTHOR]

In this paper, the simplest equation method is implemented to construct exact traveling-wave solutions to the modified Fornberg-Whitham equation. With the help of Mathematica symbolic computa- tion software, some new classes of explicit exact solutions are derived. Graphical representations of some obtained solutions are displayed. The method demonstrates the applicability, reliability and efficiency in treating a wide range of nonlinear evolution equations. [ABSTRACT FROM AUTHOR]

Abstract This paper studies the decay properties of solutions around the traveling wave solutions for Cauchy problem of the second-order Camassa–Holm equation. Applying the extended pseudo-conformal transformation methods, appearing the relevant works on the generalized Korteweg–de Vries equation (KdV) and nonlinear Schrödinger equation (NLS) from Martel and Merle, the solution is controlled by the decaying function with exponential speed, corresponding to the initial data and its second derivative with exponential decay. [ABSTRACT FROM AUTHOR]

The bifurcation method of dynamical system is applied to study traveling waves of the Joseph—Egri equation. The phase space geometry of traveling wave system of the Joseph—Egri equation is investigated in detail. We obtain the parameter bifurcation sets in which various bounded and unbounded orbits are identified and simulated. Furthermore, by the calculation of complicated elliptic integrals, exact expressions of all traveling wave solutions of the Joseph—Egri equation are given, including bounded and unbounded ones. [ABSTRACT FROM AUTHOR]

This paper is concerned with the traveling wave solutions for a discrete SIR epidemic model with a saturated incidence rate. We show that the existence and non-existence of the traveling wave solutions are determined by the basic reproduction number R 0 of the corresponding ordinary differential system and the minimal wave speed c ∗ . More specifically, we first prove the existence of the traveling wave solutions for R 0 > 1 and c > c ∗ via considering a truncated initial value problem and using the Schauder's fixed point theorem. The existence of the traveling wave solutions with speed c = c ∗ is then proved by using a limiting argument. The main difficulty is to show that the limit of a decreasing sequence of the traveling wave solutions with super-critical speeds is non-trivial. Finally, the non-existence of the traveling wave solutions for R 0 > 1 , 0 < c < c ∗ and R 0 ≤ 1 , c > 0 is proved. [ABSTRACT FROM AUTHOR]

Abstract In this contribution, theoretical model of the induced vibration and energy harvesting performances of a piezoelectric tubular energy harvester is established and analyzed. Analytical solutions to the model are obtained with the help of a system of linear operator equations. A series of simulations are done to investigate the influences of mechanical and electrical load parameters upon output performances of the energy harvester. Analyses of the characteristic roots show possible way to simplify the problem with asymptotic method. Results for state variables in the piezoelectric tube show good linear dependence of elastic displacements upon radial coordinates. Simulations on output performances indicate that the external connected electrical circuits should be compatible with the intended working conditions of the piezoelectric tubular energy harvester. The model developed here is expect to be useful in the design for self-powered sensor networks in hydraulic piping systems. [ABSTRACT FROM AUTHOR]

Abstract This paper is on study of traveling wave solutions and asymptotic spreading of a class of time periodic diffusion equations with degenerate nonlinearity. The asymptotic behavior of traveling wave solutions is investigated by using auxiliary equations and a limit process. In addition, the monotonicity and uniqueness, up to translation, of traveling wave solution with critical speed are determined by sliding method. Finally, combining super and sub-solutions and the stability of steady states, some sufficient conditions on asymptotic spreading are given, which indicates that the success or failure of asymptotic spreading are dependent on the degeneracy of nonlinearity as well as the size of compact support of initial value. [ABSTRACT FROM AUTHOR]

The main object of this paper is to study the bifurcation, chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion. A traveling wave transformation is used to simplified the perturbed stochastic nonlinear Schrödinger equation into ordinary differential equation. The dynamic behavior of two-dimensional planar dynamical systems and their perturbed systems are studied, and bifurcation, phase portrait, and Poincaré section are presented. Furthermore, traveling wave solutions included Jacobian function solutions, trigonometric function solutions and hyperbolic function solutions are constructed. • The main purpose of this paper is to study the chaotic pattern and traveling wave solution of the perturbed stochastic nonlinear Schrödinger equation with generalized anti-cubic law nonlinearity and spatio-temporal dispersion. • The dynamic behavior of two-dimensional planar dynamical system and their perturbed system are studied by using the theory of dynamics system. • Traveling wave solutions included Jacobian function solutions trigonometric function solutions and hyperbolic function solutions are constructed. [ABSTRACT FROM AUTHOR]

In this paper, we analyze spreading speed and traveling wave solutions of a reaction–diffusion Zika model with constant recruitment. By using the basic reproduction number R 0 of the corresponding ordinary differential system and the minimal wave speed c ∗ , the spreading properties of the solution of the model are established. More precisely, if R 0 < 1 , then the solution of the system converges to the disease-free equilibrium as t → ∞ and if R 0 > 1 , the entire solution of the system is uniformly persistent with ‖ x ‖ = c t , ∀ c ∈ [ 0 , c ∗) and the infectious disease gradually disappears with | x | ⩾ c t for any c > c ∗ . On the basis of it, we then analyze the full information about the existence and nonexistence of traveling wave solutions of the system involved with R 0 and c ∗. Finally, some numerical experiments are presented to modeling some conclusions. [ABSTRACT FROM AUTHOR]

This article investigates the traveling wave solution of the parabolic law nonlinear stochastic dispersive Schrödinger-Hirota equation with multiplicative white noise, and its conclusions can be applied to optical communication. Firstly, the equation is transformed into a nonlinear ordinary differential equation through traveling wave transformation, and its planar dynamic system is studied. The phase portraits of the equation in various cases are drawn by Maple software. Secondly, the traveling wave solutions of this equation are obtained by using the complete discriminant system theory of cubic polynomials. Finally, we draw two-dimensional graphics, three-dimensional graphics, density plots and contour plots for the module length of some solutions under specific parameters. In addition, the above research results can help readers have a more comprehensive and clear understanding of this equation and its application. • The paper studies the traveling wave solution and dynamics analysis of the parabolic law nonlinear stochastic dispersive Schrödinger-Hirota equation with multiplicative white noise. • The planar dynamics system of the Schrödinger-Hirota equation is discussed. • Three-dimension graphic, two-dimensional graphic, density plot and contour plot of some obtained solutions are plotted. [ABSTRACT FROM AUTHOR]

In this paper, the traveling wave solutions of the model of magneto-optic waveguide with anti-cubic nonlinearity are obtained by using the complete discrimination system for polynomial method, including singular solutions, solitary wave solutions,and double periodic solutions. And under specific parameter conditions, three types of optical wave patterns are obtained to visualize the model and demonstrate their accurate physical behavior. Then the dynamic properties of Schrödinger equation in magneto-optic waveguide with anti-cubic nonlinearity are analyzed, the existence of periodic and solitary solutions is proved based on the bifurcation method. Also the Hamiltonian properties and the classification of its equilibrium points are obtained. In final, we analyze the chaotic behavior of the model under some external perturbations. • This paper use the traveling wave transformation to convert the equation into a dynamic system and obtain its Hamiltonian properties. • Based on the bifurcation method, the existence of cycles and solitary solutions is proved. • The equation is analyzed using the complete discrimination system for polynomial method, and all its traveling wave solutions are obtained. • The external perturbation terms, in order to analyze the chaotic behavior of this equation. [ABSTRACT FROM AUTHOR]

In this work, the modified simple equation (MSE) method is applied to some class of nonlinear PDEs, namely, a system of nonlinear PDEs, a (2 + 1)-dimensional nonlinear model generated by the Jaulent–Miodek hierarchy, and a generalized KdV equation with two power nonlinearities. As a result, exact traveling wave solutions involving parameters have been obtained for the considered nonlinear equations in a concise manner. When these parameters are chosen as special values, the solitary wave solutions are derived. It is shown that the proposed technique provides a more powerful mathematical tool for constructing exact solutions for a broad variety of nonlinear PDEs in mathematical physics.

This paper is concerned with the stability of an age-structured susceptible−exposed− infective−recovered−susceptible (SEIRS) model with time delay. Firstly, the traveling wave solution of system can be obtained by using the method of characteristic. The existence and uniqueness of the continuous traveling wave solution is investigated under some hypotheses. Moreover, the age-structured SEIRS system is reduced to the nonlinear autonomous system of delay ODE using some insignificant simplifications. It is studied that the dimensionless indexes for the existence of one disease-free equilibrium point and one endemic equilibrium point of the model. Furthermore, the local stability for the disease-free equilibrium point and the endemic equilibrium point of the infection-induced disease model is established. Finally, some numerical simulations were carried out to illustrate our theoretical results.

The present paper is concerned with the spatial spreading speeds and traveling wave solutions of cooperative systems in space-time periodic habitats with nonlocal dispersal. It is assumed that the trivial solution u = 0 of such a system is unstable and the system has a stable space-time periodic positive solution u∗(t, x). We first show that in any direction ξ ∈ SN -1, such a system has a finite spreading speed interval, and under certain condition, the spreading speed interval is a singleton set, and hence, the system has a single spreading speed c∗(ξ) in the direction of ξ. Next, we show that for any c > c∗(ξ), there are space-time periodic traveling wave solutions of the form u(t, x) = Φ(x - ctξ, t, ctξ) connecting u∗ and 0, and propagating in the direction of ξ with speed c, where Φ(x, t, y) is periodic in t and y, and there is no such solution for c < c∗(ξ). We also prove the continuity and uniqueness of space-time periodic traveling wave solutions when the reaction term is strictly sub-homogeneous. Finally, we apply the above results to nonlocal monostable equations and two-species competitive systems with nonlocal dispersal and space-time periodicity. [ABSTRACT FROM AUTHOR]

In this study, we investigate the nonlinear time-fractional Korteweg–de Vries (KdV) equation by using the (1/G ′ )-expansion method and the finite forward difference method. We first obtain the exact wave solutions of the nonlinear time-fractional KdV equation. In addition, we used the finite-forward difference method to obtain numerical solutions in this equations. When these solutions are obtained, the indexed forms of both Caputo and conformable derivatives are used. By using indexing technique, it is shown that the numerical results of the nonlinear time-fractional KdV equation approaches the exact solution. The two- and three-dimensional surfaces of the obtained analytical solutions are plotted. The von Neumann stability analysis of the used numerical scheme with the studied equation is carried out. The L2 and L ∞ error norms are computed. The exact solutions and numerical approximations are compared by supporting with graphical plots and tables. [ABSTRACT FROM AUTHOR]

A spatially nonlocal diffusion model with a class of delayed nonlocal responses was considered. The asymptotic stability and the convergence rate of the traveling wavefronts were mainly studied. Through construction of weighted functions and establishment of a comparison principle for the related linear equations, the conclusion that if the initial function is within a bounded distance from a certain traveling wavefront with respect to a weighted maximum norm, the solution satisfying the initial value will converge to the traveling wavefront exponentially in time, was proved, and the exponential convergence rate was also obtained. [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]

Abstract In this paper, we investigate a degenerate reaction–diffusion/reaction–aggregation–diffusion model with non-smooth reaction term. The strong and weak properties of travelling wave front solutions for this model are analyzed. Under degenerate reaction–diffusion/reaction–aggregation–diffusion term and non-smooth reaction term, we obtain how strong and weak properties change. [ABSTRACT FROM AUTHOR]

The stability of traveling wave solutions to a class of Lotka-Volterra competitive sys­tems with age structures was studied. In the case of quasi-monotonicity, the existence and com­parison theorems for the solutions to the initial value problems of the systems were first estab­lished on R with the analytic semigroup theory and the abstract functional differential equations. Then based on the weighted energy method, the comparison theorem as well as the embedding theorem, the global exponential stability of the monostable large-speed traveling wave solutions under the so-called large initial perturbation ( i. e. the initial perturbation around the traveling wave decaying exponentially as ж ^^ , but being arbitrarily large at other locations) was ob­tained for the systems in the weighted Sobolev space. The results show that, as the steady state solution of the system, the traveling wave solution usually determines the long-term asymptotic behavior of the solution to the initial value problem. Its stability reveals that the phenomena and results of inter-species competition systems can be clearly observed without interference by ex­ternal factors. [ABSTRACT FROM AUTHOR]

We consider a time-periodic competition system with a free boundary, which can be used to describe two species that are struggling on the free boundary to obtain their own habitats and the final competition result is the regional partition of two species. In [15] , we have studied the (classical) traveling wave solutions of the temporal homogeneous problems. In this paper, we consider the time-periodic case and study periodic traveling wave solutions of the problem. More precisely, we will prove the existence and uniqueness of the periodic traveling wave solution, and show that the speed of the periodic traveling wave is uniquely determined by and monotonically dependent on the time-periodic terms in the system. Moreover, we will present some sufficient conditions to guarantee that the wave travels with a positive speed. [ABSTRACT FROM AUTHOR]

The current paper is devoted to the study of traveling wave solutions of the following parabolic–parabolicchemotaxis system, u t = Δ u − χ ∇ ⋅ ( u ∇ v ) + u ( a − b u ) , x ∈ R N τ v t = Δ v − v + u , x ∈ R N , where u ( x , t ) represents the population density of a mobile species and v ( x , t ) represents the population density of a chemoattractant, and χ represents the chemotaxis sensitivity. In an earlier work (Rachidi et al., 2017) by the authors of the current paper, traveling wave solutions of the above chemotaxis system with τ = 0 are studied. It is shown in Rachidi et al. (2017) that for every 0 < χ < b 2 , there is c ∗ ( χ ) such that for every c > c ∗ ( χ ) and ξ ∈ S N − 1 , the system has a traveling wave solution ( u ( x , t ) , v ( x , t ) ) = ( U ( x ⋅ ξ − c t ; τ ) , V ( x ⋅ ξ − c t ; τ ) ) with speed c connecting the constant solutions ( a b , a b ) and ( 0 , 0 ) . Moreover, lim χ → 0 + c ∗ ( χ ) = 2 a if 0 < a ≤ 1 1 + a if a > 1 . We prove in the current paper that for every τ > 0 , there is 0 < χ τ ∗ < b 2 such that for every 0 < χ < χ τ ∗ , there exist two positive numbers c ∗ ∗ ( χ , τ ) > c ∗ ( χ , τ ) ≥ 2 a satisfying that for every c ∈ ( c ∗ ( χ , τ ) , c ∗ ∗ ( χ , τ ) ) and ξ ∈ S N − 1 , the system has a traveling wave solution ( u ( x , t ) , v ( x , t ) ) = ( U ( x ⋅ ξ − c t ; τ ) , V ( x ⋅ ξ − c t ; τ ) ) with speed c connecting the constant solutions ( a b , a b ) and ( 0 , 0 ) , and it does not have such traveling wave solutions of speed less than 2 a . Moreover, lim χ → 0 + c ∗ ∗ ( χ , τ ) = ∞ , lim χ → 0 + c ∗ ( χ , τ ) = 2 a if 0 < a ≤ 1 + τ a ( 1 − τ ) + 1 + τ a ( 1 − τ ) + + a ( 1 − τ ) + 1 + τ a if a ≥ 1 + τ a ( 1 − τ ) + , and lim x → ∞ U ( x ; τ ) e − μ x = 1 , where μ is the only solution of the equation μ + a μ = c in the interval ( 0 , min { a , 1 + τ a ( 1 − τ ) + } ) . Furthermore, lim τ → 0 + χ τ ∗ = b 2 , lim τ → 0 + c ∗ ( χ ; τ ) = c ∗ ( χ ) , lim τ → 0 + c ∗ ∗ ( χ ; τ ) = ∞ . [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of advection hyperbolicparabolic systems with nonlocal delay. We prove that the wave profile is described by a hybrid system that consists of an integral transformation and an ordinary differential equation. By considering the same problem for a properly parameterized system and the continuous dependence of the wave speed on the parameter involved, we obtain the existence and uniqueness of traveling wave solutions in advection hyperbolic-parabolic system with nonlocal delay under bistable assumption. The inuence of advection on the propagation speed is also considered. [ABSTRACT FROM AUTHOR]