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Diffusion mechanisms in natural sciences and innovation management involve partial differential equations (PDEs). This is due to their spatio-temporal dimensions. Functional semi-discretized PDEs (with lattice spatial structures or time delays) may be even more adapted to real world problems. In the modeling process, PDEs can also formalize behaviors, such as the logistic growth of populations with migration, and the adopters’ dynamics of new products in innovation models. In biology, these events are related to variations in the environment, population densities and overcrowding, migration and spreading of humans, animals, plants and other cells and organisms. In chemical reactions, molecules of different species interact locally and diffuse. In the management of new technologies, the diffusion processes of innovations in the marketplace (e.g., the mobile phone) are a major subject. These innovation diffusion models refer mainly to epidemic models. This contribution introduces that modeling process by using PDEs and reviews the essential features of the dynamics and control in biological, chemical and new technology transfer. This paper is essentially user-oriented with basic nonlinear evolution equations, delay PDEs, several analytical and numerical methods for solving, different solutions, and with the use of mathematical packages, notebooks and codes. The computations are carried out by using the software Wolfram Mathematica®7, and C++ codes.

This article concerns the over-determined system of partial differential equations $$ Big(frac{k}{f}Big)_x+Big(frac{f}{k}Big)_y=0, quad frac{f_{y}}{k}=frac{k_x}{f},quad Big(frac{f_y}{k}Big)_y+Big(frac{k_x}{f}Big)_x=-varepsilon fk,. $$ It was shown in [6, Theorem 8.1] that this system with $varepsilon=0$ admits traveling wave solutions as well as non-traveling wave solutions. In this article we solve completely this system when $varepsilone 0$. Our main result states that this system admits only traveling wave solutions, whenever $varepsilon e 0$.

Analytical solutions are the most effective tools for understanding phenomena modeled with differential equations. On the other hand, analytical solvers may be limited in their ability to handle complex problems and we are forced to rely on approximate and numerical methods. Through the investigation of two types of fractional operators, this article aims to investigate both approaches to studying the interactive dynamics of an eco-epidemiological diffusion model. We consider two differential operators in our main model: local beta time and non-local Riesz space fractional derivative. Firstly, we propose novel classes of soliton wave solutions for the local time fractional version of the system using two efficient analytical methods. In the second part of the paper, we consider the case where the second-order space derivative operator in the problem is of the type of Riesz fractional operator that is non-local. A finite difference scheme is the main idea behind discretizing this fractional system. Throughout this article, we highlight the validity and efficiency of both analytical and numerical perspectives. The results can be examined by epidemiologists for a more comprehensive interpretation. • Two fractional operators are considered in an eco-epidemiological model. • Exact solutions to the model with local β -derivative are found via two analytical techniques. • We also investigate approximate solutions to the model with a nonlocal fractional operator. • Numerical simulations corresponding to the obtained results are provided. • The acquired results are new and have not been obtained for the model in previous papers. [ABSTRACT FROM AUTHOR]

This research aims to look into alternative unique solitary wave solutions to the perturbed Chen–Lee–Liu (CLL) equation to explain the kinetic and physical aspects of an optical fiber pulse. The perturbed CLL equation is one of the most well-known icon models derived from the well-known Schrödinger equation. Two different analytical methodologies are used to develop novel solitary wave solutions. These responses are then thoroughly evaluated using a well-known numerical approach to establish their underlying veracity. Various graphics are used to show the pulse wave analysis process and outcomes in an optical cable. We show how scientifically unique the study is by explaining the comparison figure that was made from our data and has been used in many recent research studies • Study of perturbed Chen–Lee–Liu equation. • Analytical and numerical novel constructed results. • Investigating the obtained results' accuracy. • Explaining the obtained solutions through some distinct types of sketches. [ABSTRACT FROM AUTHOR]

We study a diffusive predator–prey model of Lotka–Volterra type functional response in which both species obey the logistic growth such that the carrying capacity of the predator is proportional to the prey population and the one for prey is a constant. Both continuous and discrete diffusion are addressed. Our aim is to see whether both species can survive eventually, if an alien invading predator is introduced to the habitat of an existing prey. The answer to this question is positive under certain restriction on the parameter. Applying Schauder's fixed point theory with the help of suitable upper and lower solutions, the existence of traveling wave solutions for this model is proven. Furthermore, by deriving the non-existence of traveling wave solutions, we also determine the minimal speed of traveling waves for this model. This provides an estimation of the invasion speed. [ABSTRACT FROM AUTHOR]

Background: Recently, nonlinear time-delayed evolution equations have received considerable interest due to their numerous applications in the areas of physics, biology, chemistry and so on. Methods: In this paper, we obtain traveling wave solutions by using the extended $$\left( \frac{G^{\prime}}{G}\right)$$ -expansion method. Results: Based on the method, we get many solutions of the time-delayed generalized Burgers-type equations. Conclusions: The results reveal that the extended $$\left( \frac{G^{\prime}}{G}\right)$$ -expansion method is direct, effective and can be used for many other nonlinear time-delayed evolution equations. [ABSTRACT FROM AUTHOR]

In this article, the two variables (G'/G, 1/G)-expansion method is suggested to obtain abundant closed form wave solutions to the perturbed nonlinear Schrodinger equation and the cubic-quintic Ginzburg-Landau equation arising in the analysis of various problems in mathematical physics. The wave solutions are expressed in terms of hyperbolic function, the trigonometric function, and the rational functions. The method can be considered as the generalization of the familiar (G'/G)-expansion method established by Wang et al. The approach of this method is simple, standard, and computerized. It is also powerful, reliable, and effective. [ABSTRACT FROM AUTHOR]

The family of the tangent hyperbolic function methods is one of the most powerful method to find the solutions of the nonlinear partial differential equations. In the mathematical literature, there are a great deal of tanh-methods completing each other. In this article, the unified tanh-function method as a unification of the family of tangent hyperbolic function methods is introduced and implemented to find traveling wave solutions for three important physical models, namely the Khoklov-Zabolotskaya-Kuznetsov (KZK) equation, the Newell-Whitehead-Segel (NWS) equation, and the Rabinovich wave equation with nonlinear damping. Various exact traveling wave solutions of these physical structures are formally derived. [ABSTRACT FROM AUTHOR]

This paper uses Lie symmetry group method to study a special kind of PDE. By using the Lie symmetry analysis, all of the geometric vector fields of the equation are obtained; the symmetry reductions are also presented. Some new nonlinear wave solutions, involving differentiable arbitrary functions are obtained. [ABSTRACT FROM AUTHOR]

In order to gain a better understanding of the behavior of finite-amplitude acoustic waves under a Brinkman-based poroacoustic model, we make use of approximations and transformations to recast our model equation into the damped Burgers equation. We examine two special case solutions of the damped Burgers equation: the approximate solution to the damped Burgers equation and the boundary value problem given an initial sinusoidal signal. We study the effects of varying the Darcy coefficient, Reynolds number, and coefficient of nonlinearity on these solutions. [ABSTRACT FROM AUTHOR]

In this paper, the first integral method and a variety of solitary waves methods are employed for constructing the new exact traveling wave solutions of a (2 + 1)-dimensional Zakharov-like soliton equation. Moreover, this equation will be studied by a variety of solitary waves methods and ansatze. [ABSTRACT FROM AUTHOR]

In this paper, we present the fractal complex transform via a local fractional derivative. The traveling wave solutions for the fractal Korteweg-de Vries equations within local fractional derivative are obtained based on the special functions defined on Cantor sets. The technology is a powerful tool for solving the local fractional non-linear partial differential equations. [ABSTRACT FROM AUTHOR]

The problem of two-dimensional arbitrary amplitude low-frequency electrostatic oscillation in a quasi-neutral quantum plasma is solved exactly by elementary means. In such quantum plasmas we have treated electrons quantum mechanically and ions classically. The exact analytical solution of the nonlinear system exhibits the formation of dark and black solitons. Numerical simulation also predicts the possible periodic solution of the nonlinear system. Nonlinear analysis reveals that the system does have a bifurcation at a critical Mach number that depends on the angle of propagation of the wave. The small-amplitude limit leads to the formation of weakly nonlinear Kadomstev–Petviashvili solitons. [ABSTRACT FROM AUTHOR]

In reaction-diffusion models describing the interaction between the invading grey squirrel and the established red squirrel in Britain, Okubo et al. ([19]) found that in certain parameter regimes, the profiles of the two species in a wave propagation solution can be determined via a solution of the KPP equation. Motivated by their result, we employ an elementary approach based on the maximum principle for elliptic inequalities coupled with estimates of a total density of the three species to establish the nonexistence of traveling wave solutions for Lotka-Volterra systems of three competing species. Applying our estimates to the May-Leonard model, we obtain upper and lower bounds for the total density of a solution to this system. For the existence of traveling wave solutions to the Lotka-Volterra three-species competing system, we find new semi-exact solutions by virtue of functions other than hyperbolic tangent functions, which are used in constructing one-hump exact traveling wave solutions in [2]. Moreover, new two-hump semi-exact traveling wave solutions different from the ones found in [1] are constructed. [ABSTRACT FROM AUTHOR]

The rapid development of the Internet, especially the emergence of the social networks, leads rumor propagation into a new media era. Rumor propagation in social networks has brought new challenges to network security and social stability. This paper, based on partial differential equations (PDEs), proposes a new SIS rumor propagation model by considering the effect of the communication between the different rumor infected users on rumor propagation. The stabilities of a nonrumor equilibrium point and a rumor-spreading equilibrium point are discussed by linearization technique and the upper and lower solutions method, and the existence of a traveling wave solution is established by the cross-iteration scheme accompanied by the technique of upper and lower solutions and Schauder's fixed point theorem. Furthermore, we add the time delay to rumor propagation and deduce the conditions of Hopf bifurcation and stability switches for the rumor-spreading equilibrium point by taking the time delay as the bifurcation parameter. Finally, numerical simulations are performed to illustrate the theoretical results. [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]

In this paper, we study the spreading speed and traveling wave solutions of a non-quasi-monotone delayed reaction-diffusion model for a single species population with separate mobile and stationary states. By using comparison arguments, Schauder’s fixed-point theorem and a limiting process, we establish the existence of the spreading speed and characterize it as the minimal wave speed for traveling wave solutions. The upward convergence of the spreading speed and traveling wave solutions are also established by applying a fluctuation method. In particular, the effects of the delay and transfer rates on the spreading speed are investigated. [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. [ABSTRACT FROM AUTHOR]

We apply the qualitative theory of dynamical systems to study exact solutions and the dynamics of quadratic and cubic nonlinear oscillators with damping. Under certain parametric conditions, we also consider the van der Waals normal form, Chaffee-Infante equation, compound Burgers-KdV equation and Burgers-KdV equation for explicit representations of kink-profile wave solutions and unbounded traveling wave solutions. [ABSTRACT FROM AUTHOR]

We extend our previous work on the spatial spread of phage infection of immobile bacteria on an agar coated plate by explicitly including loss of viruses by both adsorption to bacteria and by decay of free viruses and by including a distributed virus latent period and distributed burst size rather than fixed values of these key parameters. We extend earlier results on the spread of virus and on the existence of traveling wave solutions when the basic reproductive number for virus, R0, exceeds one and we compare the results with those obtained in earlier work. Finally, we formulate and analyze a model of multiple virus strains competing to infect a common bacterial host in a petri dish. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of traveling wave solution to a three species reaction-diffusion system with delays, which includes competitive relationship, cooperative relationship and predator-prey relationship. By using the method of upper-lower solutions, the cross iteration method and Schauder's fixed point theorem, the existence of a traveling wave solution is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of non-cooperative diffusion–reaction systems, which include prey–predator models and disease-transmission models. The concept of weak traveling wave solutions is proposed. The necessary and sufficient conditions for the existence of such solutions are obtained by the Schauder's fixed-point theorem and persistence theory. The introduction of persistence theory is very technical and crucial. The LaSalle's invariance principle is applied to show that traveling wave solutions connect two equilibria. The nonexistence of traveling wave solutions is proved by introducing a negative one-sided Laplace transform. The results are applied to a prey–predator model and a disease-transmission model with specific interaction functions. We find that the profile of traveling wave solutions may depend on different eigenvalues according to the corresponding condition, which is a new phenomenon. [ABSTRACT FROM AUTHOR]

In this study, we consider exact traveling wave solutions of the Bogoyavlenskii equation using a modified extended tanh-function method. This method has a broad applicability to many other nonlinear evolution equations in mathematical physics. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of traveling wave fronts for a generalized KdV–mKdV equation. We first establish the existence of traveling wave solutions for the equation without delay, and then we prove the existence of traveling wave fronts for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using geometric singular perturbation theory, Fredholm theory and the linear chain trick.

In this paper, we approximate traveling wave solutions via artificial neural networks. Finding traveling wave solutions can be interpreted as a forward-inverse problem that solves a differential equation without knowing the exact speed. In general, we require additional restrictions to ensure the uniqueness of traveling wave solutions that satisfy boundary and initial conditions. This paper is based on the theoretical results that the bistable three-species competition system has a unique traveling wave solution on the premise of the monotonicity of the solution. Since the original monotonic neural networks are not smooth functions, they are not suitable for representing solutions of differential equations. We propose a method of approximating a monotone solution via a neural network representing a primitive function of another positive function. In the numerical integration, the operator learning-based neural network resolved the issue of differentiability by replacing the quadrature rule. We also provide theoretical results that a small training loss implies a convergence to a real solution. The set of functions neural networks can represent is dense in the solution space, so the results suggest the convergence of neural networks with appropriate training. We validate that the proposed method works successfully for the cases where the wave speed is identical to zero. Our monotonic neural network achieves a small error, suggesting that an accurate speed and solution can be estimated when the sign of wave speed is known.

In this paper, we discuss the existence and asymptotic behavior of traveling waves for a generalized Burgers-KdV equation. We show the heteroclinic orbits of the associated ordinary differential equations for the generalized Burgers-KdV equation with a special convolution kernel and then establish the existence result of traveling wave solutions for the Burgers-KdV equation by employing geometric singular perturbation theory and the linear chain trick. And the asymptotic behavior of traveling waves is obtained by using the standard asymptotic theory. [ABSTRACT FROM AUTHOR]

This paper is concerned with periodic traveling wave solutions of the forced generalized nearly concentric Korteweg-de Vries equation in the form of (uη+u/(2η)+[f(u)]ξ+uξξξ)ξ+uθθ/η2=h0. The authors first convert this equation into a forced generalized Kadomtsev-Petviashvili equation, (ut+[f(u)]x+uxxx)x+uyy=h0, and then to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relationship between the ordinary differential equation and nonlinear integral equations with symmetric kernels is established by using the Green's function method. The integral representations generate compact operators in a Banach space of real-valued continuous functions. The Schauder's fixed point theorem is then used to prove the existence of nonconstant solutions to the integral equations. Therefore, the existence of periodic traveling wave solutions to the forced generalized KP equation, and hence the nearly concentric KdV equation, is proved.

Various motions of camphor boats in the water channel exhibit both a homogeneous and an inhomogeneous state, depending on the number of boats, when unidirectional motion along an annular water channel can be observed even with only one camphor boat. In a theoretical research, the unidirectional motion is represented by a traveling wave solution in a model. Since the experimental results described above are thought of as a kind of bifurcation phenomena, we would like to investigate a linearized eigenvalue problem in order to prove the destabilization of a traveling wave solution. However, the eigenvalue problem is too diffcult to analyze even if the number of camphor boats is 2. Hence we need to make a reduction on the model. In the present paper, we apply the center manifold theory and reduce the model to an ordinary differential system. [ABSTRACT FROM AUTHOR]

The Riccati-Bernoulli sub-ODE method is firstly proposed to construct exact traveling wave solutions, solitary wave solutions, and peaked wave solutions for nonlinear partial differential equations. A Bäcklund transformation of the Riccati-Bernoulli equation is given. By using a traveling wave transformation and the Riccati-Bernoulli equation, nonlinear partial differential equations can be converted into a set of algebraic equations. Exact solutions of nonlinear partial differential equations can be obtained by solving a set of algebraic equations. By applying the Riccati-Bernoulli sub-ODE method to the Eckhaus equation, the nonlinear fractional Klein-Gordon equation, the generalized Ostrovsky equation, and the generalized Zakharov-Kuznetsov-Burgers equation, traveling solutions, solitary wave solutions, and peaked wave solutions are obtained directly. Applying a Bäcklund transformation of the Riccati-Bernoulli equation, an infinite sequence of solutions of the above equations is obtained. The proposed method provides a powerful and simple mathematical tool for solving some nonlinear partial differential equations in mathematical physics. [ABSTRACT FROM AUTHOR]

This paper is concerned with the wave length λ of smooth periodic traveling wave solutions of the Camassa–Holm equation. The set of these solutions can be parametrized using the wave height a (or “peak-to-peak amplitude”). Our main result establishes monotonicity properties of the map a ⟼ λ ( a ) , i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated with the equation, which distinguish between the two possible qualitative behaviors of λ ( a ) , namely monotonicity and unimodality. The key point is to relate λ ( a ) to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems. [ABSTRACT FROM AUTHOR]

The homogeneous balance of undetermined coefficients method is proposed to obtain not only exact solutions but also multi-symplectic structure of some nonlinear partial differential equations. Bilinear equation, N-soliton solutions, traveling wave solutions and multi-symplectic structure are obtained by applying the proposed method to the KdV equation. Accordingly, the definition and multi-symplectic structure of the generalized KdV-type equation are given. The proposed method is also a standard and computable method, which can be generalized to deal with some types of nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

This paper is devoted to discussing bifurcation and traveling wave solutions for the Fokas equation. By investigating the dynamical behavior with phase space analysis, we may derive all possible exact traveling wave solutions, including compactons, cuspons, periodic cusp wave solutions, and smooth solitary wave solutions. [ABSTRACT FROM AUTHOR]

The traveling wave solution of a unified higher-order traffic flow model is investigated with a discontinuous fundamental diagram under the Lagrange coordinate. The equilibrium velocity is a piecewise function which consists of two concave functions. The weak solution theory is applied to study the traveling wave solution of the model, in which a set of equations about the characteristic parameters are obtained. Through numerical simulation, the moving cluster solutions of the anisotropic and isotropic traffic flow models are reproduced, respectively. The numerical results agree with the analytical ones. [ABSTRACT FROM AUTHOR]

An approximate analytical solution of Richards' equation (RE) in a semi-finite layer of porous medium is obtained in this article. The basic idea is constructing a function between the Boltzmann variable ϕ and the traveling wave solution of RE, introducing the function into RE and calculating the relevant parameters by differential transform method (DTM). In the end, three examples are prepared to show the accuracy of the presented approximate analytical solution. [ABSTRACT FROM AUTHOR]

We study a cooperative system of integro-differential equations. It is shown that the system in general has multiple spreading speeds, and when the linear determinacy conditions are satisfied all the spreading speeds are the same and equal to the spreading speed of the linearized system. The existence of traveling wave solutions is established via integral systems. It is shown that when the linear determinacy conditions are satisfied, if the unique spreading speed is not zero then it may be characterized as the slowest speed of a class of traveling wave solutions. Some examples are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We establish the existence and non-existence of traveling wave solutions for a diffusive SIR model with a general nonlinear incidence. The existence proof is shown by introducing an auxiliary system, applying Schauder’s fixed point theorem and then a limiting argument. The nonexistence proof is obtained by two-sided Laplace transform when the speed is less than the critical velocity. Numerical simulations support the theoretical results. We also point out the effects of the delay and the diffusion rate of the infective individuals on the spreading speed. [ABSTRACT FROM AUTHOR]

In this paper, we show the existence and qualitative properties of traveling wave solutions to the Allen–Cahn equation with fractional Laplacians. A key ingredient is the estimation of the traveling speed of traveling wave solutions. [ABSTRACT FROM AUTHOR]