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 paper we review the existence of diierent types of travelling wave solutions u(x; t) = (x ? ct) of degenerate non-linear reaction-diiusion equations of the form u t = D(u)u x ] x + g(u) for diierent density-dependent diiusion coeecients D and kinetic part g. These include the non-linear degenerate generalized Fisher-KPP and the Nagumo equations. Also, we consider an equation whose diius...

An extended ( ′ G )–expansion method is obtained by improving the form of solution in ( G′ G )– expansion method which is proposed in recent years. By using the extended ( ′ G )–expansion method and with the aid of homogeneous balance principle, many explicit and exact travelling wave solutions with two arbitrary parameters to the Klein-Gordon equation are presented, including the hyperbolic so...

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...

In this paper, we establish exact solutions for nonlinear evolution equations. The homogeneous balance method is used to construct exact travelling wave solutions of the CalogeroDegasperis (CD) and Potential Kadomstev-Petviashvili (PKP) equations, in which the homogeneous balance method is applied to solve the Riccati equation and the reduced nonlinear ordinary differential equation. Many new f...

In this paper, new travelling wave solutions to the Fornberg-Whitham equation ut − uxxt + ux + uux = uuxxx + 3uxuxx are investigated. They are characterized by two parameters. The expresssions of the periodic and solitary wave solutions are obtained.

We study entropy travelling wave solutions for rst order hyperbolic balance laws. Results concerning existence, regularity and asymptotic stability of such solutions are proved for convex uxes and source terms with simple isolated zeros.

By using the method of dynamical systems, this paper researches the bifurcation and the exact travelling wave solutions for (2 + 1)-dimensional nonlinear models generated by the Jaulent-Miodek hierarchy. Exact parametric representations of all wave solutions are given.

In this paper we investigate the existence of travelling wave solutions for a chemotaxis model under the scenarios of zero growth and constant growth rate. We use Lie symmetry analysis to generate generalized travelling wave solutions, a wider class of solutions than that obtained from the standard ansatz. Unlike previous approaches, we allow for diffusivity and signal degradation. We study the...

A two-dimensional micro-scale model for crystal dissolution and precipitation in a porous medium is presented. The local geometry of the pore is represented as a thin strip. The model allows for changes in the pore volume. A formal limiting argument leads to a system of 1D effective upscaled equations. The effective equations allow for travelling wave solutions. Existence and uniqueness of thes...