This paper concentrates on a particular first-order coupled PDE system. It provides both a detailed treatment of the existence and uniqueness of monotone travelling waves to various equilibria, by differential-equation theory and by probability theory and a treatment of the corresponding hyperbolic initial-value problem, by analytic methods. The initial-value problem is studied using characteri...

A Hamiltonian formulation of nonlinear, parallel propagating, travelling whistler waves is developed. The complete system of equations reduces to two coupled differential equations for the transverse electron speed u and a phase variable φ=φp − φe representing the difference in the phases of the transverse complex velocities of the protons and the electrons. Two integrals of the equations are o...

In this paper, the modified Korteweg-de Vries (mKdV) equation with variable coefficients (vc-mKdV equation) is investigated via two kinds of approaches and symbolic computation. On the one hand, we firstly reduce the vc-mKdV equation to a second-order nonlinear nonhomogeneous ODE using travelling wave-like similarity transformation. And then we obtain its many types of exact fractional solution...

The travelling wave dielectrophoretic pump studied here is essentially a rectangular straight micro-channel with an electrode array on part of its wall, and operated under an ac voltage with phase shift at neighbouring electrodes. The travelling wave dielectrophoretic force drives the cells, which drag the plasma, and after some sophisticated interaction between conventional dielectrophoresis, ...

We consider bistable reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. The discrete diffusion term is allowed to have periodic or even negative coefficients. We show that travelling wave solutions to such pure lattice systems exist and that they can be approximated by travelling wave solutions to a system that incorporates both local and non-local diffu...

In this article, the (G ′ /G)-expansion method has been implemented to find the travelling wave solutions of nonlinear evolution equations of fractional order. For this, the fractional complex transformation method has been used to convert fractional order partial differential equation to ordinary differential equation. Then, (G ′ /G)-expansion method has been implemented to celebrate the serie...

The existence of travelling wave solutions for a density-dependent selection migration model in population genetics is proven. A single locus and two alleles are assumed. It is also assumed that the fitnesses of the heterozygotes in the population are below those of the homozygotes. The method of proof is by constructing an isolating neighborhood and computing a connection index. Key words, pop...

A generalized and improved G′/G -expansion method is proposed for finding more general type and new travelling wave solutions of nonlinear evolution equations. To illustrate the novelty and advantage of the proposed method, we solve the KdV equation, the Zakharov-KuznetsovBenjamin-Bona-Mahony ZKBBM equation and the strain wave equation in microstructured solids. Abundant exact travelling wave s...

We consider atomic chains with nearest neighbour interactions and study periodic and homoclinic travelling waves which are called wave trains and solitons, respectively. Our main result is a new existence proof which relies on the constrained maximisation of the potential energy and exploits the invariance properties of an improvement operator. The approach is restricted to convex interaction p...