Explicit Traveling Wave Solutions Research Articles

Generalized travelling wave solutions for hyperbolic chemotaxis PDEs

The chemotaxis of E. coli can be modelled via a system of partial differential equations. We apply group theoretical techniques to investigate the existence of travelling wave solutions for the hyperbolic chemotaxis model. Unlike previous approaches, we allow for the presence of the diffusivity terms, thereby maintaining the intrinsic second order terms of the diffusion equations in the model. We show, in the case of non starvation, that this model is still tractable. Explicit travelling wave solutions are determined.

Comment on “Exact explicit travelling wave solutions for (n+1)-dimensional Klein–Gordon–Zakharov equations” [Chaos, solitons and fractals 34(2007) 867–871

Jibin Li studied “exact explicit travelling wave solutions for (n+1)-dimensional Klein–Gordon–Zakharov equations.” By using the approach of dynamical system, the author claimed that they had obtained five classes of exact explicit travelling wave solutions. However, we checked the results and found two typographical technical errors in the main results. Furthermore, the author fails to note, for a=0,b<0,h∈(0,∞) and a<0,b>0,h=0, there is a family of periodic wave solutions and a series of breaking wave solutions, respectively.

Kink and kink-like waves in pre-stretched Mooney-Rivlin viscoelastic rods

The present paper theoretically investigates kink and kink-like waves propagating in pre-stretched Mooney-Rivlin viscoelastic rods. In the constitutive modeling, the Cauchy stress tensor is assumed to consist of an elastic part and a dissipative part. The asymptotic method is adopted to simplify the nonlinear dynamic equations in the limit of finite-small amplitude and long wavelength. Using the reductive perturbation method, we further derive the well-known far-field equation (i.e. the KdV-Burgers equation), to which two kinds of explicit traveling wave solutions are presented. Examples are given to show the influences of pre-stretch and viscosity on the wave shape and wave velocity. It is shown that pre-stretch could be an effective method for modulating the two types of waves. In addition, such waves may be utilized to measure the viscosity coefficient of the material. The competition between the effects of pre-stretch and viscosity on the kink and kink-like waves is also revealed.

Bifurcations and exact solutions for a nonlinear surface wind waves equation

By using the method of dynamical systems, for the nonlinear surface wind waves equation, which is given by Manna, we study its dynamical behavior to determine all exact explicit traveling wave solutions. To guarantee the existence of the aforementioned solutions, all parameter conditions are determined. Our procedure shows that the nonlinear surface wind waves equation has no peakon solution. Copyright © 2015 John Wiley & Sons, Ltd.

Asymptotically stable states of nonlinear age-structured monocyclic population model I. Travelling wave solution

This paper is devoted to the study of evolutionary dynamics of monocyclic age-structured population including effect of nonlinear mortality (population growth feedback) and proliferation. The total population is considered as partitioned by fixing age into two subpopulations. Individuals of first population are born, mature, die and can at the final fixed age give birth for some new individuals (with null age). Individuals of the second subpopulation are older than those of the first one. They can mature, die and do not have possibility to proliferate. This model was considered as a system of two initial–boundary value problems for nonlinear transport equations with non-local boundary conditions. We obtained explicit travelling wave solution provided the model parameters (coefficients of equations and initial values) satisfy the restrictions that guarantee continuity and smoothness of solution. Explicit form of solution allowed us to perform numerical experiments with high accuracy using the set of parameterized algebraic functions which do not depend from the time. In all performed experiments solutions are attracted to some stationary functions for a long time period (asymptotically stable states of system). We indicated and studied three different regimes of population dynamics. The first is quasi-equilibrium regime, when the maximum value of population density by age, as a function of time, is attracted to the point from the neighbourhood of initial value. This is a result of balance between effects of proliferation and nonlinear mortality (like behaviour of microorganism population in the cases of asymptomatic or healthy carriers). The second and third regimes are characterized by increasing (decreasing) maximum by age values of population density with following attracting to values higher (lower) than initial one. We studied also the impact of parameters of nonlinear death rate on the tremendous growth of population density followed by transition to asymptotically stable states (like infection generalization process in living organisms).

Travelling Wave Solution and Recurrent Algorithm for the Nonlinear Age-Structured Model of Polycyclic Population Dynamics

This work is devoted to the study of evolutionary dynamics of polycyclic age-structured one-sex population. It extends previous work on polycyclic population including the effect of nonlinear mortality (population growth feedback) and proliferation. Further, the population is assumed to have two types of individuals, proliferating and quiescent. The resulting mathematical model is based on the systems of initial-boundary value problems for the non-linear transport equation with integral boundary condition. This model describes both the temporal and age evolution of the density of the proliferating and quiescent subpopulations. Using the method of characteristics and some modification of the steps method we derive a multistep recurrent algorithm which produces explicit travelling wave solutions. The compatibility conditions impose the restrictions on the model parameters and guarantee continuity and continuously differentiable of obtained solution. The explicit form of this solution is used to create the accurate recurrent numerical algorithm and carry out the series of numerical experiments with a set of parameterized algebraic functions. We study the asymptotical behavior of numerical solutions for the periodically oscillating functions of model parameters and obtained the quasi-periodical regimes of population dynamics in the form of harmonic oscillations and pulse sequence in the series of experiments.

New exact traveling wave solutions to the (1+1)-dimensional Klein-Gordon-Zakharov equation for wave propagation in plasma using the exp(-Φ(ξ))-expansion method

The (1+1)-dimensional nonlinear Klein-Gordon-Zakharov equation considered as a model equation for describing the interaction of the Langmuir wave and the ion acoustic wave in high frequency plasma. By the execution of the exp(-Φ(ξ))-expansion, we obtain new explicit and exact traveling wave solutions to this equation. The obtained solutions include kink, singular kink, periodic wave solutions, soliton solutions and solitary wave solutions of bell types. The variety of structure and graphical representation make the dynamics of the equations visible and provides the mathematical foundation in plasma physics and engineering.

An exponential expansion method and its application to the strain wave equation in microstructured solids

Abstract The modeling of wave propagation in microstructured materials should be able to account for various scales of microstructure. Based on the proposed new exponential expansion method, we obtained the multiple explicit and exact traveling wave solutions of the strain wave equation for describing different types of wave propagation in microstructured solids. The solutions obtained in this paper include the solitary wave solutions of topological kink, singular kink, non-topological bell type solutions, solitons, compacton, cuspon, periodic solutions, and solitary wave solutions of rational functions. It is shown that the new exponential method, with the help of symbolic computation, provides an effective and straightforward mathematical tool for solving nonlinear evolution equations arising in mathematical physics and engineering.

Analytical Treatment of the Evolutionary (1 + 1)-Dimensional Combined KdV-mKdV Equation via the Novel (G'/G)-Expansion Method

The novel (G'/G)-expansion method is a powerful and simple technique for finding exact traveling wave solutions to nonlinear evolution equations (NLEEs). In this article, we study explicit exact traveling wave solutions for the (1 + 1)-dimensional combined KdV-mKdV equation by using the novel (G'/G)-expansion method. Consequently, various traveling wave solutions patterns including solitary wave solutions, periodic solutions, and kinks are detected and exhibited.

Classification of Multiply Travelling Wave Solutions for Coupled Burgers, Combined KdV-Modified KdV, and Schrödinger-KdV Equations

Some explicit travelling wave solutions to constructing exact solutions of nonlinear partial differential equations of mathematical physics are presented. By applying a theory of Frobenius decompositions and, more precisely, by using a transformation method to the coupled Burgers, combined Korteweg-de Vries- (KdV-) modified KdV and Schrödinger-KdV equation is written as bilinear ordinary differential equations and two solutions to describing nonlinear interaction of travelling waves are generated. The properties of the multiple travelling wave solutions are shown by some figures. All solutions are stable and have applications in physics.

NOTES ON EXACT TRAVELLING WAVE SOLUTIONS FOR A LONG WAVE-SHORT WAVE MODEL

This paper considers a long wave-short wave model. It shows that under three different parameter conditions, this system has three types of exact explicit travelling wave solutions. Their parametric representations have been given.

Observations on the class of “Balancing Principle” for nonlinear PDEs that can be treated by the auxiliary equation method

There is collection of methods for finding explicit travelling wave solutions of nonlinear partial differential equations (NLPDEs) in the literature. Quite a large amount of these methods employ “Balancing Principle” in their methodology and they work for positive integer values only. Yet there is no well established formula for balancing and there is no specific rule to determine adequate balancing principle that works for any number which is positive/or negative and/or rational. In this study, we propose a new balancing principle which works for large range of the numbers either positive/or negative and/or rational that makes it possible to obtain new and original explicit travelling wave solutions of nonlinear partial differential equations (NLPDEs).

Bifurcations and Exact Traveling Wave Solutions for a Model of Nonlinear Pulse Propagation in Optical Fibers

In this paper, we consider a model of nonlinear pulse propagation in optical fibers. By investigating the dynamical behavior and bifurcations of solutions of the traveling wave system of PDE, we derive all possible exact explicit traveling wave solutions under different parameter conditions. These results completed the study of traveling wave solutions for the mentioned model posed by [Lenells, 2009].

Explicit periodic travelling waves for a discrete lambda–omega reaction–diffusion system

In 1973, Kopell and Howard introduced a λ–ω reaction–diffusion system and found an explicit family of periodic travelling wave solutions lying on circles with radius less than 1. Since λ–ω systems represent universal models for studying chemical processes, and onset of turbulent behaviour, etc., explicit solutions of λ–ω systems with delays or discrete λ–ω systems can be of further help when the only method for obtaining other solutions is through numerical computation. There are now much investigations of various λ–ω systems. However, it is of interest to note that none attempts to find explicit travelling wave solutions. In this paper, we investigate the existence of explicit solutions for the simplest Euler scheme of a λ–ω system with delays or advancements which is described as a coupled pair of partial difference equations. We are able to provide necessary as well as sufficient conditions for the existence of numerical periodic travelling wave solutions. Additionally, we also provide some examples to show that our explicit solutions are qualitatively different from those found by Kopell and Howard and hence they may be of interests for specialists in the area of reaction–diffusion systems.

Exact traveling wave solutions of some nonlinear evolution equations

Using a traveling wave reduction technique, we have shown that Maccari equation, (2?1)-dimensional nonlinear Schrodinger equation, medium equal width equation, (3?1)-dimensional modified KdV-Zakharov- Kuznetsev equation, (2?1)-dimensional long wave-short wave resonance interaction equation, perturbed nonlinear Schrodinger equation can be reduced to the same family of auxiliary elliptic-like equations. Then using extended F-expansion and projective Riccati equation methods, many types of exact traveling wave solutions are obtained. With the aid of solutions of the elliptic-like equation, more explicit traveling wave solutions expressed by the hyper- bolic functions, trigonometric functions and rational func- tions are found out. It is shown that these methods provide a powerful mathematical tool for solving nonlinear evolu- tion equations in mathematical physics. A variety of structures of the exact solutions of the elliptic-like equation are illustrated.

New Exact Traveling Wave Solutions for Some Coupled BBM Equations

The present paper deals with results of explicit traveling wave solutions for some coupled BBM equations. By detailed computation and using the ′   G G -expansion method, many traveling wave solutions are given. These traveling waves are in the form of hyperbolic functions, the trigonometric functions and the rational functions, which show the reliability and efficiency of the used method.

Transition from pulled to pushed premixed turbulent flames due to countergradient transport

The influence of countergradient transport on the speed of a statistically stationary, planar, 1D premixed flame that propagates in frozen turbulence is studied theoretically and numerically by considering the normalised magnitude NB of the countergradient flux to be an input parameter. Spectra of admissible flame speeds are analytically determined and explicit travelling wave solutions are found for two algebraic relations widely used to close the mean rate of product creation. A problem of selecting the physically relevant solution that is approached for sufficiently steep initial conditions is addressed. It is argued that, if NB is larger than an analytically determined critical number NcrB, then the type of the physically relevant solution is drastically changed. If NB < NcrB, the physically relevant solution is of pulled wave type, i.e. its speed is controlled by processes localised to the leading edge of the flame brush and can be determined within the framework of a linear analysis at the leading edge. If NB > NcrB, the physically relevant solution is of pushed wave type, i.e. its speed is controlled by processes in the entire flame brush. Analytical expressions for the speed of the physically relevant solution as a function of NB and the density ratio are obtained. For NB > NcrB, the mean flame brush thickness and the spatial profile of the Favre-averaged combustion progress variable are also determined analytically. These results are validated by numerical simulations. Both analytical expressions and numerical data indicate that (i) both turbulent flame speed and thickness are decreased when NB is increased and (ii) the direction of total scalar flux (i.e. the sum of countergradient and gradient contributions) is strongly affected not only by NB, but also by the shape of the dependence of the mean rate of product creation on the mean combustion progress variable.

Symmetry reductions, dynamical behavior and exact explicit solutions to the Gordon types of equations

In this paper, the combination of Lie symmetry analysis and the dynamical system method is performed on the mixed second-order sine-Gordon equation, all of the geometric vector fields of the sine-Gordon equation, the generalized nonlinear wave equation and its special case, the Liouville equation are presented. Then, the symmetry reductions and exact solutions to such nonlinear wave equations are considered. Especially, the bifurcations of the sine-Gordon equation are obtained, and the exact explicit traveling wave solutions are investigated by the dynamical system method. To guarantee the existence of the traveling wave solutions, all of the parameter conditions are determined.

Existence of periodic travelling wave solutions of non-autonomous reaction–diffusion equations with lambda–omega type

Periodic travelling wave solutions of reaction–diffusion equations were studied by many authors. The λ–ω type reaction–diffusion system is a notable special model that admits explicit periodic travelling wave solutions and was introduced by Kopell and Howard in 1973. There are now similar systems which are investigated by means of autonomous dynamics. In contrast, there are few papers which are concerned with non-autonomous cases. For this reason, we apply Mawhin’s continuation theorem to derive the existence of periodic travelling wave solutions for non-autonomous λ–ω systems, and we describe the ‘disappearance’ of periodic travelling wave solutions under special situations. Our main result is also illustrated by examples.

Qualitative analysis and explicit traveling wave solutions for the Davey–Stewartson equation

In this paper, we use the bifurcation method of dynamical systems to study the traveling wave solutions for the Davey–Stewartson equation. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow‐up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended. Copyright © 2013 John Wiley &amp; Sons, Ltd.