In this work, we extended the application of ”the modified reductive perturbation method” to symmetrical regularized long waves with quadratic nonlinearity and obtained various form of KdV equations as the governing equations. Seeking a localized travelling wave solutions to these evolution equations we determined the scale parameters g1 and g2 so as to remove the possible secularities that mig...

We discuss travelling-wave solutions of a system of coupled reaction-diffusion equations used by the authors to describe the macroscopic behaviour of fungal mycelia. Such systems have been used in a multitude of applications; and, in particular, Merkin and Needham (1990, Proc. R. Soc. Lond. A 430, 315-45) have studied a certain formulation as a model for generic isothermal chemical reactions. W...

In this paper the travelling wave solutions in the adiabatic model with two-step chain branching reaction mechanism are investigated both numerically and analytically in the limit of equal diffusivity of reactant, radicals and heat. The properties of these solutions and their stability are investigated in detail. The behaviour of combustion waves are demonstrated to have similarities with the p...

We consider a coupled system of Hamiltonian partial differential equations introduced by Popowicz, which has the appearance of a two-field coupling between the Camassa-Holm and Degasperis-Procesi equations. The latter equations are both known to be integrable, and admit peaked soliton (peakon) solutions with discontinuous derivatives at the peaks. A combination of a reciprocal transformation wi...

In this paper we present a model for crystal dissolution in porous media and we analyse travelling wave solutions of the ensuing equations for a one{dimensional ow situation. We demonstrate the structure of the waves and we prove existence and uniqueness. The travelling wave description is characterized by a rate parameter k and a diiusion/dispersion parameter D. We investigate the limit proces...

In this paper, we establish exact solutions for a nonlinear evolution equation. The sech method and the exp-function method are used to construct the solitary travelling wave solutions of Lax’s seventh-order KdV equation. These solutions may be important of significance for the explanation of some practical physical problem. Crown Copyright 2008 Published by Elsevier Inc. All rights reserved.

An application of the generalized tanh–coth method to search for exact solutions of nonlinear partial differential equations is analyzed. This method is used for variants of the KdV–Burger and the K(n, n)–Burger equations. The generalized tanh–coth method was used to construct periodic wave and solitary wave solutions of nonlinear evolution equations. This method is developed for searching exac...

We investigate the existence and the asymptotic stability of travelling wave solutions for a hyperbolic 2 2 system with a relaxation source term. Using the subcharacteristic stability condition, which implies special monotonicity properties of the solutions, we are able to establish the L 1 asymptotic attractivity of these solutions. In this paper we investigate nonlinear stability of travellin...

In this paper, the first integral method is proposed to solve the KolmogorovPetrovskii-Piskunov equation. New exact travelling wave solutions of the KolmogorovPetrovskii-Piskunov equation are obtained that illustrate the efficiency of the method. 2010 Mathematics Subject Classification: 35K57

Lie symmetry group method is applied to study the Fisher-Kolmogorov equation. The symmetry group is given, and travelling wave solutions are obtained. Finally the conservation laws are determined.