Propagation and Pinning of Travelling Wave for Nagumo Type Equation

In this paper, we study the propagation and its failure to propagate (pinning) of a travelling wave in a Nagumo type equation, an equation that describes impulse propagation in nerve axons that also models population growth with Allee effect. An analytical solution is derived for the traveling wave and the work is extended to a discrete formulation with a piecewise linear reaction function. We propose an operator splitting numerical scheme to solve the equation and demonstrate that the wave either propagates or gets pinned based on how the spatial mesh is chosen.

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS FOR THE GENERALIZED DODD-BULLOUGH-MIKHAILOV EQUATION

In this paper, the generalized Dodd-Bullough-Mikhailov equation is studied. The existence of periodic wave and unbounded wave solutions is proved by using the method of bifurcation theory of dynamical systems. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.Some exact explicit parametric representations of the above travelling solutions are obtained.

Study on Atmospheric Travelling Wave Solutions and Review of Its Present Developments

The scientific achievements of travelling waves in a barotropic atmosphere are introduced, including i) the existence conditions of periodic solutions (wavetrain solutions) and solitary wave solutions (pulse solutions), together with the solution finding methods and a series of related problems, ii) seeking solutions of monotonous wave (wave front) and of nonmonotonous travelling wave (oscillatory wave) by using phase plane shooting technique and hi) progress in the study of travelling wave solution at home and abroad. The investigation of travelling wave solutions in recent years has been found in mathematics, physics, chemistry, biology and other sciences. Over the past decade the problem has been the subject of much interest and become an important area of research. So it is no doubt of great significance to investigate the travelling wave solutions and thereby explain phenomena of weather.

THE SMOOTH AND NONSMOOTH TRAVELLING WAVESOLUTIONS IN A NONLINEAR WAVE EQUATION

The travelling wave solutions (TWS) in a class of P. D. E. is studied. The travelling wave equation of this P. D. E. is a planar cubic polynomial system in three-parameter space. The study for M became the topological classifications of bifurcations of phase portraits defined by the planar system. B using the theory of planar dynamical systems to do qualitative analysis, all topological classifications of the cubic polynomial system can be obtained. Returning the results of the phase plane analysis to TWS, u(xi), and considering discontinuity of the right side of the equation of TWS when xi = x - ct is varied along a phase orbit and passing through a singular curve, all conditions of existence of smooth and nonsmooth travelling waves are given.

Exact travelling wave solutions for （1＋ 1）-dimensional dispersive long wave equation

A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.

ASYMPTOTIC METHOD OF TRAVELLING WAVE SOLUTIONS FOR A CLASS OF NONLINEAR REACTION DIFFUSION EQUATION

In this article the travelling wave solution for a class of nonlinear reaction diffusion problems are considered. Using the homotopic method and the theory of travelling wave transform, the approximate solution for the corresponding problem is obtained.

The classification of travelling wave solutions and superposition of multi-solutions to Camassa-Holm equation with dispersion

Under the travelling wave transformation, the Camassa-Holm equation with dispersion is reduced to an integrable ordinary differential equation （ODE）, whose general solution can be obtained using the trick of one-parameter group. Furthermore, by using a complete discrimination system for polynomial, the classification of all single travelling wave solutions to the Camassa-Holm equation with dispersion is obtained. In particular, an affine subspace structure in the set of the solutions of the reduced ODE is obtained. More generally, an implicit linear structure in the Camassa-Holm equation with dispersion is found. According to the linear structure, we obtain the superposition of multi-solutions to Camassa-Holm equation with dispersion.

The (G'/G, 1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations

The （G＇/G, 1/G）-expansion method for finding exact travelling wave solutions of nonlinear evolution equations, which can be thought of as an extension of the （G＇/G）-expansion method proposed recently, is presented. By using this method abundant travelling wave so- lutions with arbitrary parameters of the Zakharov equations are successfully obtained. When the parameters are replaced by special values, the well-known solitary wave solutions of the equations are rediscovered from the travelling waves.

TRAVELLING WAVE SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS BY USING SYMBOLIC COMPUTATION

A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the parameter can be applied in judging the existence of various forms of travelling wave solutions.An efficiency of this method is demonstrated on some equations,which include Burgers Huxley equation,Caudrey Dodd Gibbon Kawada equation,generalized Benjamin Bona Mahony equation and generalized Fisher equation.

A new auxiliary equation method for finding travelling wave solutions to KdV equation

In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the （1＋1）-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.

THE NONLINEAR STABILITY OF TRAVELLING WAVE SOLUTIONS FOR A REACTING FLOW MODEL WITH SOURCE TERM

In this paper, author considers a 3 x 3 system for a reacting flow model propesed by [9]. Since this model has source term, it can be considered as a relaxation approximation to 2 x 2 systems of conservation laws, which include the well-known p-system. From this viewpoint, the author establishes the global existence and the nonlinear stability of travelling wave solutions by L-2 energy method.

CONVERGENCE RATES TO TRAVELLING WAVES FOR A RELAXATION MODEL WITH LARGE INITIAL DISTURBANCE

This paper is concerned with the convergence rates to travelling waves for a relaxation model with general flux functions. Compared with former results in this direction, the main novelty in this paper lies in the fact that the initial disturbance can be chosen large in suitable norm. Our analysis is based on the L^1-stability results obtained by C. Mascia and R. Natalini in [12].

NEW EXACT TRAVELLING WAVE SOLUTIONS TO THREE NONLINEAR EVOLUTION EQUATIONS

Based on the computerized symbolic computation, some new exact travelling wave solutions to three nonlinear evolution equations are explicitly obtained by replacing the tanhξ in tanh-function method with the solutions of a new auxiliary ordinary differential equation.

Analysis on influence of seismic travelling wave effect on semi-active control for long-span rigid-continuous bridge

The analysis approach of semi-active control for long-span rigid-continuous bridge under seismic travelling wave input is established. Magnetorheological dampers are set on the positions of the bridge beatings. The semi-active control calculation and analysis are performed for a five-span rigid-continuous bridge under seismic travelling waves with different apparent surface velocities. The results indicate that travelling wave effect remarkably influences the uncontrolled seismic responses, the semi-active control seismic responses and vibration control effects for the long-span rigid-continuous bridge. It is disadvantageous to the responses of the beams and the piers under the travelling wave input with lower apparent surface velocity, and travelling wave effect can decrease the vibration control effects evidently. Therefore, the travelling wave effect should be considered for the selection of the parameter values of semi-active control system in order to get the designing control effect.

Travelling Waves in a Compressible Mooney-Rivlin Rod

This article deals with waves in a circular cylindrical rod composed of a compressible Mooney Rivlin material. All kinds of travelling waves and the conditions for their existence are studied by using the method of nonlinear dynamics.

Exact Travelling Wave Solutions for a Combined KdV and Schwarzian KdV Equation

In this paper the ( G'/G )-expansion method is used to find exact travelling wave solutions for a combined KdV and Schwarzian KdV equation. As a result, multiple travelling wave solutions with arbitrary parameters are obtained, which are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are derived from the travelling waves. The (G'/G)-expansion method presents a wider applicability for handling nonlinear wave equations.

Travelling Wave in the Generalized Kuramoto Model with Inertia

We study the dynamics of the generalized Kuramoto model with inertia, in which oscillators with positive cou- pling strength are conformists and oscillators with negative coupling strength are contrarians. By numerically simulating the model, we find that the model supports a modulated travelling wave state except for already displayed travelling wave states and π state in previous be characterized by the phase distributions of oscillators. travelling wave state of the model in the parameter space literature. The modulated travelling wave state may Finally, the modulated travelling wave state and the are presented.

Theoretical research on the performance of a practical helix travelling wave tube

On the basis of a rigorous field theory, two different physical models of attenuator and sever have been proposed. One is named High attenuation （HATT） model in which both attenuator and sever are considered as a unified attenuator, but the sever is regarded as an area of very high loss; the other is called Sever and attenuator （SATT） model in which the sever is modelled as a drift area in which the electric and magnetic fields both vanish. A complex function is derived and potential sinking effect is also considered. Thus, a set of more practical self-consistent equations of nonlinear beam-wave interaction is formulated. Simulations are carried out under the conditions of the two different physical models, and the simulation results are compared with the experimental data. The results show that in the case of single signal drive, the unknown second harmonic should be included for predicting the saturated output power. It is also evident that the SATT model and the HATT model predict the same physical nature, whereas the results predicted by the HATT model are much closer to the experimental data than those obtained from the SATT model. Therefore, these results provide a strong theoretical basis for designing broadband and high gain helix travelling wave tubes.

Ocean internal waves interpreted as oscillation travelling waves in consideration of ocean dissipation

Most studies of the synthetic aperture radar remote sensing of ocean internal waves are based on the solitary wave solutions of the Korteweg-de Vries （KdV） equation, and the dissipative term in the KdV equation is not taken into account. However, the dissipative term is very important, both in the synthetic aperture radar images and in ocean models. In this paper, the traveling-wave structure to characterize the ocean internal wave phenomenon is modeled, the results of numerical experiments are advanced, and a theoretical hypothesis of the traveling wave to retrieve the ocean internal wave parameters in the synthetic aperture radar images is introduced.

Analysis of the axial electric field in a plasma-loaded- helix travelling wave tube

A helix type slow wave structure filled with plasma is immersed in a strong longitudinal magnetic field. Taking into account the effect of the plasma and the dielectric, the system is separated radially into three regions. By means of the sheath model and Maxwell equation, the distribution of the electromagnetic field is established. Using the boundary conditions of each region, the dispersion relation of the slow wave structure is derived. The trend of change for the radial profile of the axial electric field is analysed respectively in different plasma densities, plasma column radius and dielectric constant by numerical computation. Some useful results are obtained on the basis of the discussion.