Travelling Waves for a Density Dependent Diffusion Nagumo Equation over the Real Line

We consider the density dependent diffusion Nagumo equation, where the diffusion coefficient is a simple power function. This equation is used in modelling electrical pulse propagation in nerve axons and in population genetics (amongst other areas). In the present paper, the δ-expansion method is applied to a travelling wave reduction of the problem, so that we may obtain globally valid perturbation solutions (in the sense that the perturbation solutions are valid over the entire infinite domain, not just locally; hence the results are a generalization of the local solutions considered recently in the literature). The resulting boundary value problem is solved on the real line subject to conditions at z → ±∞. Whenever a perturbative method is applied, it is important to discuss the accuracy and convergence properties of the resulting perturbation expansions. We compare our results with those of two different numerical methods (designed for initial and boundary value problems, respectively) and deduce that the perturbation expansions agree with the numerical results after a reasonable number of iterations. Finally, we are able to discuss the influence of the wave speed c and the asymptotic concentration value α on the obtained solutions. Upon recasting the density dependent diffusion Nagumo equation as a two-dimensional dynamical system, we are also able to discuss the influence of the nonlinear density dependence (which is governed by a power-law parameter m) on oscillations of the travelling wave solutions.     

Travelling Waves for the Gross-Pitaevskii Equation II

The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results (see [3,4,8]) where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy.     

A Non-Existence Result for Supersonic Travelling Waves in the Gross-Pitaevskii Equation

We prove the non-existence of non-constant travelling waves of finite energy and of speed in the Gross-Pitaevskii equation in dimension N2.     

On Travelling Waves for the Stochastic Fisher–Kolmogorov–Petrovsky–Piscunov Equation

This paper is concerned with properties of the wave speed for the stochastically perturbed Fisher–Kolmogorov–Petrovsky–Piscunov (FKPP) equation. It was shown in the classical 1937 paper by Kolmogorov, Petrovsky and Piscunov that the large time behavior of the solution to the FKPP equation with Heaviside initial data is a travelling wave. In a seminal 1995 paper Mueller and Sowers proved that this also holds for a stochastically perturbed FKPP equation. The wave speed depends on the strength σ of the noise. In this paper bounds on the asymptotic behavior of the wave speed c(σ) as σ→0 and σ→∞ are obtained.     

Fractional Fokker-Planck Equation with Space and Time Dependent Drift and Diffusion

In this paper, we aim to answer the question proposed by Magdziarz (Stoch. Proc. Appl. 119:3238?C3252, 2009), i.e. we investigate the solution of an anomalous diffusion equation with time and space dependent force and diffusion coefficient. First, we try to find the stochastic representation of this equation, which means the PDF of this stochastic process is rightly the solution of the equation we aim to solve. Then, we also simulate the sample paths of the stochastic process. At last, taking advantage of the stochastic representation method, we employed Monte Carlo method to approximate the solution of the mentioned equation.     

Similarity Reductions for a Nonlinear Diffusion Equation

Abstract

Similarity reductions and new exact solutions are obtained for a nonlinear diffusion equation. These are obtained by using the classical symmetry group and reducing the partial differential equation to various ordinary differential equations. For the equations so obtained, first integrals are deduced which consequently give rise to explicit solutions. Potential symmetries, which are realized as local symmetries of a related auxiliary system, are obtained. For some special nonlinearities new symmetry reductions and exact solutions are derived by using the nonclassical method.     

New Exact Travelling Wave Solutions for Zakharov-Kuznetsov Equation

In this paper, the nonlinear dispersive Zakharov- Kuznetsov equation is solved by using the generalized auxiliary equation method. As a result, new solitary pattern, solitary wave and singular solitary wave solutions are found.     

KdV-Burgers Equation for a Viscous Flowing Shallow Water Waves

Korteweg, de Vries-Burges equation is obtained for an incompressible and viscous fluid which is flowing in one direction for the shallow water. We assume that the wave amplitude is small but finite, the viscosity of the fluid is also small enough.     

Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation

We study various statistical properties of real roots of three different classes of random polynomials which recently attracted a vivid interest in the context of probability theory and quantum chaos. We first focus on gap probabilities on the real axis, i.e. the probability that these polynomials have no real root in a given interval. For generalized Kac polynomials, indexed by an integer d, of large degree n, one finds that the probability of no real root in the interval [0,1] decays as a power law n ^−θ(d) where θ(d)>0 is the persistence exponent of the diffusion equation with random initial conditions in spatial dimension d. For n≫1 even, the probability that they have no real root on the full real axis decays like n ^{−2(θ(2)+θ(d))}. For Weyl polynomials and Binomial polynomials, this probability decays respectively like and where θ _∞ is such that in large dimension d. We also show that the probability that such polynomials have exactly k roots on a given interval [a,b] has a scaling form given by where N _ab is the mean number of real roots in [a,b] and a universal scaling function. We develop a simple Mean Field (MF) theory reproducing qualitatively these scaling behaviors, and improve systematically this MF approach using the method of persistence with partial survival, which in some cases yields exact results. Finally, we show that the probability density function of the largest absolute value of the real roots has a universal algebraic tail with exponent −2. These analytical results are confirmed by detailed numerical computations. Some of these results were announced in a recent letter (Schehr and Majumdar in Phys. Rev. Lett. 99:060603, 2007).     

Solitary Waves of a Perturbed sine—Gordon Equation

Based on the bifurcation and the idea that the solitary waves and shock waves of partial differential equations correspond respectively to the homoclinic and heteroclinic trajectories of nonlinear ordinary differential equations satisfied by the travelling waves,different conditions for the existence of solitary waves of a perturbed sine-Gordon equation are obtained.All of the corresponding approximate solitary wave solutions are given by integrating the derived approximate equations directly.     

Travelling Solitary Wave Solutions for Generalized Time-delayed Burgers-Fisher Equation

In this paper, travelling wave solutions for the generalized time-delayed Burgers-Fisher equation are studied. By using the first-integral method, which is based on the ring theory of commutative algebra, we obtain a class of travelling solitary wavesolutions for the generalized time-delayed Burgers-Fisher equation.A minor error in the previous article is clarified.     

Interactions of Soliton Waves for a Generalized Discrete KdV Equation

It is well known that soliton interactions in discrete integrable systems often possess new properties which are different from the continuous integrable systems, e.g., we found that there are such discrete solitons in a semidiscrete integrable system(the time variable is continuous and the space one is discrete) that the shorter solitary waves travel faster than the taller ones. Very recently, this kind of soliton was also observed in a full discrete generalized Kd V system(the both of time and space variables are discrete) introduced by Kanki et al. In this paper, for the generalized discrete Kd V(gd Kd V) equation, we describe its richer structures of one-soliton solutions. The interactions of two-soliton waves to the gd Kd V equation are studied. Some new features of the soliton interactions are proposed by rigorous theoretical analysis.     

Travelling Wave Solutions for Konopelchenko-Dubrovsky Equation Using an Extended sinh-Gordon Equation Expansion Method

The sinh-Gordon equation expansion method is further extended by generalizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the (2+1)-dimensional Konopelchenko-Dubrovsky equation is investigated and abundant exact travelling wave solutions are explicitly obtained including solitary wave solutions, trigonometric function solutions and Jacobi elliptic doubly periodic function solutions, some of which are new exact solutions that we have never seen before within our knowledge. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Travelling Wave Solutions for Konopelchenko-Dubrovsky Equation Using an Extended sinh-Gordon Equation Expansion Method

The sinh-Gordon equation expansion method is further extended by generMizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the （2＋1）-dimensional Konopelchenko-Dubrovsky equation is investigated and abundant exact travelling wave solutions are explicitly obtained including solitary wave solutions, trigonometric function solutions and Jacobi elliptic doubly periodic function solutions, some of which are new exact solutions that we have never seen before within our knowledge. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Travelling Solitary Wave Solutions for Generalized Time-delayed Burgers-Fisher Equation

In this paper, travelling wave solutions for the generalized time-delayed Burgers-Fisher equation are studied. By using the first-integral method, which is based on the ring theory of commutative algebra, we obtain a class of travelling solitary wave solutions for the generalized time-delayed Burgers-Fisher equation. A minor error in the previous article is clarified.     

The Classical Diffusion Equation for a Fully Ionized Plasma-Revisited

Very simple, but very accurate approximate solutions are given for the spatial part of the classical diffusion equation for a fully ionized plasma.     

Exact Travelling Wave Solutions to a Coupled Nonlinear Evolution Equation

By using an improved hyperbola function method, several types of exact travelling wave solutions to a coupled nonlinear evolution equation are obtained, which include kink-shaped soliton solutions, bell-shaped soliton solutions, envelop solitary wave solutions, and new solitary waves. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Nonsingular Travelling Complexiton Solutions to a Coupled Korteweg--de Vries Equation

A class of novel nonsingular travelling complexiton solutions to a coupled Korteweg-de Vries （KdV） equation is presented via the first step Darboux transformation of the complex KdV equation with nonzero seed solution. Furthermore, the properties of the nonsingular solutions are discussed.     

Exact Travelling Wave Solutions to a Coupled Nonlinear Evolution Equation

By using an improved hyperbola function method, several types of exact travelling wave solutions to a coupled nonlinear evolution equation are obtained, which include kink-shaped soliton solutions, bell-shaped soliton solutions, envelop solitary wave solutions, and new solitary waves. The method can be applied to other nonlinear evolution equations in mathematical physics.