We establish the classical wave equation for a particle formed of a mass-less oscillatory elementary charge generally also traveling, and the resulting electromagnetic waves, of a generally Doppler-effected angular frequency ω, in the vacuum in three dimensions. We obtain from the solutions the total energy of the particle wave to be ε = ¯ h c ω, 2π¯ h c being a function expressed in wave-mediu...

For a class of scalar partial differential equations that incorporate convection, diffusion, and possibly dispersion in one space and one time dimension, the stability of traveling wave solutions is investigated. If the initial perturbation of the traveling wave profile decays at an algebraic rate, then the solution is shown to converge to a shifted wave profile at a corresponding temporal alge...

Although the modified simple equation (MSE) method effectively provides exact traveling wave solutions to nonlinear evolution equations (NLEEs) in the field of engineering and mathematical physics, it has some limitations. When the balance number is greater than one, usually the method does not give any solution. In this article, we have exposed a process how to implement the MSE method to solv...

A simple system is studied, involving a single nondispersive breaking wave and its interaction with two dispersive modes through a resonant triad. The dynamics of this system are shown to be quite rich, through a combined theoretical and numerical analysis. A sharply defined traveling wave with a corner seems to attract almost all initial data with enough energy, provided the nondispersive wave...

We consider the lubrication model for a thin film driven by competing gravitational forces and thermal gradients on an inclined plane. We are interested in the general traveling wave problem when the Navier slip boundary condition is used. We contrast (1) gravity dominated flow, (2) Marangoni dominated flow, and (3) flow in which the two driving effects balance. For a “singular slip” model we s...

We consider the population dynamics of two competing species sharing the same resource, which is modeled by the carrying capacity term of logistic equation. One species (farmer) increases the carrying capacity in exchange for a decreased survival rate, while the other species (exploiter) does not. As the carrying capacity is shared by both species, farmer is altruistic. The effect of continuous...

We study a singularly perturbed system of partial differential equations that models a one-dimensional array of coupled Chua’s circuits. The PDE system is a natural generalization to the FitzHugh-Nagumo equation. In part I of the paper, we show that similar to the FitzHugh-Nagumo equation, the system has periodic traveling wave solutions formed alternatively by fast and slow flows. First, asymp...

In this paper we present an intuitive explanation for the non-uniqueness of the traveling wave speed in the Fisher equation, showing a similar non-uniqueness property in the case of inviscid traveling waves. More precisely, we prove that traveling waves of the Fisher equation with wave speed c > 0 converges to the inviscid traveling wave with speed c > 0 as the diffusion vanishes. A complete di...

The principle of the traveling wave based differential protection for bipolar HVDC transmission lines is proposed in the paper. Unlike the traditional current differential protection, the quantity of current is replaced by the quantity of the traveling wave for comparison. The traveling wave at the remote end is transferred to the local end for comparison to the local traveling wave. For the bi...