We consider periodic traveling wave solutions to the focusing nonlinear Schrödinger equation (NLS) that have been shown to persist when the NLS is perturbed to the complex Ginzburg-Landau equation (CGL). In particular, we show that these periodic traveling waves are spectrally stable solutions of NLS with respect to periodic perturbations. Furthermore, we use an argument based on the Fredholm a...

Over the years, (G'/G)-expansion method is employed to generate traveling wave solutions to various wave equations in mathematical physics. In the present paper, the alternative (G'/G)-expansion method has been further modified by introducing the generalized Riccati equation to construct new exact solutions. In order to illustrate the novelty and advantages of this approach, the (1+1)-dimension...

The Kuramoto-Sivashinsky (KS) equation is known as a popular prototype to represent a system in which the transport of energy through nonlinear mode coupling produces a balance between long wavelength instability and short wavelength dissipation. Existing numerical results indicate that the KS equation admits three classes (namely, regular shock, oscillatory shock, and solitary wave) of nonperi...

The focus of this paper is on traveling wave solutions of the equation u t + f(u) x = u xx + 2 u xxx ; in which the ux function f is trilinear and nonconvex. In particular, it is shown that for combinations of parameters in certain ranges, there are traveling waves that converge as ?! 0 to undercompressive shocks, in which the characteristics pass through the shock. The analysis is based on exp...

Flux-limited Keller-Segel (FLKS) model has been recently derived from kinetic transport models for bacterial chemotaxis and shown to represent better the collective movement observed experimentally. Recently, associated to the kinetic model, a new instability formalism has been discovered related to stiff chemotactic response. This motivates our study of traveling wave and aggregation in popula...

This paper deals with entire solutions for a general nonlocal dispersal monostable equation with spatiotemporal delay, i.e., solutions that are defined in the whole space and for all time t ∈ R. We first derive a particular model for a single species and show how such systems arise from population biology. Then we construct some new types of entire solutions other than traveling wave solutions ...

Periodic traveling waves (wavetrains) have been extensively studied for reaction-diffusion equations. One important motivation for this work has been the identification of periodic traveling wave patterns in spatiotemporal data sets in ecology. However, for many ecological populations, diffusion is no more than a rough phenomenological representation of dispersal, and spatial convolution with a...

‎in this paper‎, ‎we obtain exact solutions involving parameters of some nonlinear pdes in mathmatical physics; namely the one-‎dimensional modified complex ginzburg-landau equation by using the $ (g^{'}/g) $ expansion method‎, homogeneous balance method, extended f-expansion method‎. ‎by ‎using homogeneous balance principle and the extended f-expansion, more periodic wave solutions expres...

Under the traveling wave transformation, Camassa-Holm equation with dispersion is reduced to an integrable ODE whose general solution can be obtained using the trick of one-parameter group. Furthermore combining complete discrimination system for polynomial, the classifications of all single traveling wave solutions to the Camassa-Holm equation with dispersion is obtained. In particular, an aff...

We study the existence and uniqueness of traveling wave solutions for a class of twocomponent reaction diffusion systems with one species being immobile. Such a system has a variety of applications in epidemiology, bio-reactor model, and isothermal autocatalytic chemical reaction systems. Our result not only generalizes earlier results of Ai and Huang (Proceedings of the Royal Society of Edinbu...