Approximate Traveling Wave Research Articles

Nonlinear dynamical modelling of high frequency electrostatic drift waves using fluid theoretical approach in magnetized plasma

A novel third order nonlinear evolution equation governing the dynamics of the recently observed high frequency electrostatic drift waves has been derived in the framework of a plasma fluid model in an inhomogeneous magnetized plasma. The corresponding linear dispersion relation arising out of the fluid equations has been studied in detail for the conventional low frequency as well as the recently observed high frequency electrostatic drift waves along with the electrostatic ion cyclotron waves. The derived nonlinear evolution equation of third order is then decomposed into two second order equations as the order of the equation becomes reduced after this kind of decomposition under certain conditions, and one equation becomes linear whereas the other equation becomes nonlinear. The detailed analysis of fixed points leading to bifurcations has been performed using the qualitative theory of differential equations and the bifurcation theory of planar dynamical systems for the reduced second order nonlinear equation. The bifurcation curves for the individual fixed points are also shown for the variations of different parameters in the parameter space of this reduced nonlinear equation. Then some exact as well as approximate travelling wave solutions of this reduced nonlinear equation have been derived. As the other second order reduced equation is linear in nature, its exact oscillatory and exponential solutions are obtained. The intersection of the solutions of these two reduced second order equations provides the solutions of the original third order nonlinear evolution equation; it is verified that the solutions of the reduced nonlinear second order equation are subsets of the oscillatory solution of the reduced linear second order equation in most cases whereas the intersection is different if the exponential solution of the reduced linear second order equation is considered. This further implies that the solutions of the reduced second order nonlinear equation can directly represent the solutions of the original third order nonlinear evolution equation in most cases representing the dynamics of the nonlinear high frequency electrostatic drift waves. These solutions of the novel third order nonlinear evolution equation represent certain vortex-like structures as our work is restricted to (1 + 1) dimensions only. The possible applications of our novel results on the dynamics of the high frequency electrostatic drift waves are discussed along with future directions for research in this field.

Numerical Analysis and Approximate Travelling Wave Solutions for a Higher Order Internal Wave System

In this work we focus on the numerical solution of a higher order bidirectional nonlinear model of Boussinesq type involving a nonlocal operator. Based on a von Neumann stability analysis for the linearized problem, an efficient and stable scheme for the nonlinear system is proposed. Our method is based on a numerical scheme known from the literature that solves satisfactorily a lower order linear system. Additionally, approximate periodic travelling wave solutions profiles for the higher order nonlinear system are presented. Such approximate travelling wave solutions are obtained from a solitary wave family of solutions for the Intermediate Long Wave (ILW) equation and the regularized Intermediate Long Wave (rILW) equation.

Perturbative Traveling-Wave Solution for a Flux-Limited Reaction-Diffusion Morphogenesis Equation

In this study, we investigate a porous medium-type flux limited reaction--diffusion equation that arises in morphogenesis modeling. This nonlinear partial differential equation is an extension of the generalized Fisher--Kolmogorov--Petrovsky--Piskunov (Fisher-KPP) equation in one-dimensional space. The approximate analytical traveling wave solution is found by using a perturbation method. We show that the morphogen concentration propagates as a sharp wave front where the wave speed has a saturated value. The numerical solutions of this equation are also provided to compare them with the analytical predictions. Finally, we qualitatively compare our theoretical results with those obtained in experimental studies.

Asymptotic approximations to travelling waves in the diatomic Fermi-Pasta-Ulam lattice$^\dagger$

We construct high-order approximate travelling waves solutions of the diatomic Fermi-Pasta-Ulam lattice using asymptotic techniques which are valid for arbitrary mass ratios. Separately small amplitude ansatzs are made for the motion of the lighter and heavier particles, which are coupled The Fredholm alternative is used to derive consistency conditions, whose solution generates small amplitude expansions for both sets of particles.

New analytic solutions of the space-time fractional Broer–Kaup and approximate long water wave equations

In the present paper, the exp(−ϕ(ξ)) expansion method is applied to the fractional Broer–Kaup and approximate long water wave equations. The explicit approximate traveling wave solutions are obtained by using this method. Here, fractional derivatives are defined in the conformable sense. The obtained traveling wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Simulations of the obtained solutions are given at the end of the paper.

An algorithm for the approximate solution of nonlinear total time fractional-order Burger equation

In this paper we have established the simplest equation method to find approximate solutions for total Burgurs equations with time fractional derivative. This method is effective in finding approximate traveling wave solutions of nonlinear, fractional evolution equations (NLEEs) in mathematical physics. The effectiveness of this manageable method has been shown by applying it to several examples in time fractional Burgers equations. The present approach has the potential to apply to other nonlinear fractional differential equations. All numerical calculations in the present study have been performed on a PC, applying programs written in Mathematica and Maple.

The variational iteration method for characteristic problem of strong damping generalized sine-Gordon equation

A class of nonlinear strong damping sine-Gordon disturbed evolution differential equation is studied which appears widely in mathematics and mechanics. Firstly, we introduce a traveling wave transformation, and obtain the exact solution of degenerate equation. Then a functional calculating method for variational iteration is constructed, thus an iterative expansion is found. Finally, the approximate traveling wave analytic solutions for the original strong damping generalized sine-Gordon disturbed evolution equation are found. The arbitrary order approximate solutions, and the simple variational iteration method are obtained with higher accuracy. The approximate analytic solution can make up for the imperfection of the simple numerical simulation solution.

A novel method for travelling wave solutions of fractional Whitham–Broer–Kaup, fractional modified Boussinesq and fractional approximate long wave equations in shallow water

In this paper, the analytical approximate traveling wave solutions of Whitham–Broer–Kaup (WBK) equations, which contain blow‐up solutions and periodic solutions, have been obtained by using the coupled fractional reduced differential transform method. By using this method, the solutions were calculated in the form of a generalized Taylor series with easily computable components. The convergence of the method as applied to the WBK equations is illustrated numerically as well as analytically. By using the present method, we can solve many linear and nonlinear coupled fractional differential equations. The results justify that the proposed method is also very efficient, effective and simple for obtaining approximate solutions of fractional coupled modified Boussinesq and fractional approximate long wave equations. Numerical solutions are presented graphically to show the reliability and efficiency of the method. Moreover, the results are compared with those obtained by the Adomian decomposition method (ADM) and variational iteration method (VIM), revealing that the present method is superior to others. Copyright © 2014 John Wiley & Sons, Ltd.

Kinetic Relations for a Lattice Model of Phase Transitions

The aim of this article is to analyse travelling waves for a lattice model of phase transitions, specifically the Fermi–Pasta–Ulam chain with piecewise quadratic interaction potential. First, for fixed, sufficiently large subsonic wave speeds, we rigorously prove the existence of a family of travelling wave solutions. Second, it is shown that this family of solutions gives rise to a kinetic relation which depends on the jump in the oscillatory energy in the solution tails. Third, our constructive approach provides a very good approximate travelling wave solution.

On the non-local hydrodynamic-type system and its soliton-like solutions

We consider a hydrodynamic system of balance equations for mass and momentum. This system is closed by the dynamic equation of state, taking into account the effects of spatio-temporal non-localities. Using group theory reduction, we obtain a system of ODEs, describing a set of approximate traveling wave solutions to the source system. The factorized system, containing a small parameter, proves to be Hamiltonian when the parameter is zero. Using Melnikov’s method, we show that the factorized system possesses, in general, a one-parameter family of homoclinic loops, corresponding to the approximate soliton-like solutions of the source system.

Approximate traveling wave solution for shallow water wave equation

Reconstruction of Variational Iteration Method (RVIM) is used for computing the coupled Whitham–Broer–Kaup shallow water. Then RVIM solution is verified against exact one and is compared with powerful approximate solutions, the Homotopy Perturbation Method (HPM) and Homotopy Analysis Method (HAM). The existent error of the methods is computed and convergence of the RVIM solution has been presented. Results obtained expose effectiveness and capability of this method to solve the nonlinear systems in mechanics, analytically.

EXACT AND APPROXIMATE TRAVELING WAVES OF REACTION-DIFFUSION SYSTEMS VIA A VARIATIONAL APPROACH

Reaction-diffusion systems arise in many different areas of the physical and biological sciences, and traveling wave solutions play special roles in some of these applications. In this paper, we develop a variational formulation of the existence problem for the traveling wave solution. Our main objective is to use this variational formulation to obtain exact and approximate traveling wave solutions with error estimates. As examples, we look at the Fisher equation, the Nagumo equation, and an equation with a fourth-degree nonlinearity. Also, we apply the method to the multi-component Lotka–Volterra competition-diffusion system.

APPROXIMATE TRAVELING WAVE SOLUTION OF AVIAN FLU TELEGRAPH REACTION DIFFUSION EQUATION

It is known that, the telegraph equation is more suitable than ordinary diffusion equation in modeling reaction diffusion in several branches of sciences. In this paper we generalize the governing equation of distributed-infective model which represents the spread of avian flu to the telegraph reaction diffusion equation and presents its approximate traveling wave solution by using linear piecewise approximation.

Approximate traveling wave solutions for coupled Whitham–Broer–Kaup shallow water

Homotopy Perturbation Method (HPM) was used for computing the Coupled Whitham–Broer–Kaup Shallow Water. Then HPM solution verified against exact one and compared with another approximate solution, the Homotopy Analysis Method (HAM). The existent error of the methods computed and convergence of the HPM solution has presented. Results reveal that HPM is an effective and powerful in solving the non-linear systems in mechanic, analytically.

Dissipative travelling wave solution for El Niño tropic sea-air coupled oscillator

El Nino and Southern Oscillation (ENSO) is an interannual phenomenon involved in the tropical Pacific sea-air interactions. An asymptotic method of solving equations for the ENSO model is proposed. Based on a class of oscillator of ENSO model and by employing a simple and valid method of the variational iteration, the coupled system for a sea-air oscillator model of interdecadal climate fluctuations is studied. Firstly, by introducing a set of functionals and computing the variationals, the Lagrange multipliers are obtained. And then, the generalized variational iteration expressions are constructed. Finally, by selecting appropriate initial iteration, and from the iterations expressions, the approximations of solution for the sea-air oscillator ENSO model are solved successively. The approximate dissipative travelling wave solution of equations for corresponding ENSO model is studied. It is proved from the results that the method of the variational iteration can be used for analyzing the sea surface temperature anomaly in the equatorial Pacific of the sea-air oscillator for ENSO model.

Coherent flow states in a square duct

The flow in a square duct is considered. Finite amplitude approximate traveling wave solutions, obtained using the self-sustaining-process approach introduced by Waleffe [Phys. Fluids 9, 883 (1997)], are obtained at low to moderate Reynolds numbers and used as initial conditions in direct numerical simulations. The ensuing dynamics is analyzed in a suitably defined phase space. Only one among the traveling wave solutions found is capable of surviving for a long time, with the flow trajectory forming quasiregular loops in phase space. Eventually, also this trajectory escapes along the manifold of a chaotic saddle and relaminarization ensues.

Approximate Traveling Wave Solutions of Coupled Whitham‐Broer‐Kaup Shallow Water Equations by Homotopy Analysis Method

The homotopy analysis method (HAM) is applied to obtain the approximate traveling wave solutions of the coupled Whitham‐Broer‐Kaup (WBK) equations in shallow water. Comparisons are made between the results of the proposed method and exact solutions. The results show that the homotopy analysis method is an attractive method in solving the systems of nonlinear partial differential equations.

Travelling Waves of Attached and Detached Cells in a Wound-Healing Cell Migration Assay

During a wound-healing cell migration assay experiment, cells are observed to detach and undergo mitosis before reattaching as a pair of cells on the substrate. During experiments with mice 3T3 fibroblasts, cell detachment can be confined to the wavefront region or it can occur over the whole wave profile. A multi-species continuum approach to modelling a wound-healing assay is taken to account for this phenomenon. The first cell population is composed of attached motile cells, while the second population is composed of detached immotile cells undergoing mitosis and returning to the migrating population after successful division. The first model describes cell division occurring only in the wavefront region, while a second model describes cell division over the whole of the scrape wound. The first model reverts to the Fisher equation when the reattachment rate relative to the detachment rate is large, while the second case does not revert to the Fisher equation in any limit. The models yield travelling wave solutions. The minimum wave speed is slower than the minimum Fisher wave speed and its dependence on the cell detachment and reattachment rate parameters is analysed. Approximate travelling wave profiles of the two cell populations are determined asymptotically under certain parameter regimes.

Constructing Soliton and Kink Solutions of PDE Models in Transport and Biology

We present a review of our recent works directed towards discovery of a periodic, kink-like and soliton-like travelling wave solutions within the models of transport phenomena and the mathematical biology. Analytical description of these wave patterns is carried out by means of our modification of the direct algebraic balance method. In the case when the analytical description fails, we propose to approximate invariant travelling wave solutions by means of an infinite series of exponential functions. The effectiveness of the method of approximation is demonstrated on a hyperbolic modification of Burgers equation.

Traveling waves with dispersive variability and time delay.

We first determine an approximate traveling wave profile for the Cook model [J. Murray, Mathematical Biology I: An Introduction (Springer, New York, 2002), pp. 471-478] for the case in which the number of dispersers is small relative to the number of nondispersers. The results are consistent with the previous linearized wavefront analysis that predicts, counterintuitively, that relatively few dispersers can drive the population expansion wave with a wavespeed not too different from that for the case of a single dispersing population as described by the Fisher equation. The method of solution differs from that used in the latter case since here the dimensionless wavespeed is close to unity. We next generalize the Cook model to include time-delay effects. While the Cook model, like the Fisher equation, does not adequately describe the wave of advance during the Neolithic transition in Europe, we show that the generalized Cook model provides a close agreement with the historical record.