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 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...

This paper deals with the existence of travelling wave fronts of delayed reaction diffusion systems with partial quasi-monotonicity. We propose a concept of “desirable pair of upper-lower solutions”, through which a subset can be constructed. We then apply the Schauder’s fixed point theorem to some appropriate operator in this subset to obtain the existence of the travelling wave fronts.

We consider a model for the propagation of a subsonic detonation wave through a porous medium introduced by Sivashinsky [8]. We show that it admits travelling wave solutions that converge in the limit of zero temperature diffusivity to the travelling fronts of a reduced system constructed in [6].

We study travelling wave solutions for a class of fourth order parabolic equations. Travelling wave fronts of the form u(x, t) = U(x + ct), connecting homogeneous states, are proven to exist in various cases: connections between two stable states, as well as connections between an unstable and a stable state, are considered.

We investigate the diffusive Leslie–Gower predator–prey model. Travelling wave solutions were found and a minimum wave speed relationship was derived. Linear stability analysis was performed in addition to full numerical simulation of the model. All travelling waves were found to be stable. © 2015 Elsevier Inc. All rights reserved.

In this paper we establish a method to obtain the stability of periodic travelling-wave solutions for equations of Korteweg–de Vries-type ut + uux − Mux = 0, with M being a general pseudodifferential operator and where p ≥ 1 is an integer. Our approach uses the theory of totally positive operators, the Poisson summation theorem, and the theory of Jacobi elliptic functions. In particular we obta...

The space derivatives of Freidlin's travelling wave like solutions of generalized KPP equations are considered in this paper. We give estimates of the rst two space derivatives on the wave front and show that the travelling wave is nearly at on the trough and on the crest. Diierentiation of heat semigroups, logarithmic transformation and semi-classical analysis based on stochastic analysis are ...

We propose a model for transport of solutes in a porous medium participating in a dissolu-tion/precipitation reaction, in general not in equilibrium. For an unbounded spatial domain, travelling wave solutions exist, if and only if the charge distribution is constant and we deal with a dissolution situation. The travelling wave in fact exhibits a sharp dissolution front. The wave is given in a n...