Exact Solution of Some Nonlinear Differential Equations by Hirota Method

Abstract

The Hirota Bilinear Method is applied to construct exact analytical one solitary wave solutions of some class of nonlinear differential equations. first one the system of multidimensional nonlinear wave equation with the reaction part in form of the third order polynomial determined by three distinct constant vectors. Second one is the mixed diffusion wave equation in one dimension. The bilinear representation is derived by extracting one of the vector roots (unstable in general). This allows us reduce the cubic nonlinearity to a quadratic one. In our approach, the velocity of solitary wave is xed by truncating the Hirota perturbation expansion and it is found in terms of all three roots. Furthermore, Hirota Bilinear Method is also proposed to solve Brusselator reaction model. The simulations of solutions are illustrated for diffusion wave equation in one dimension. The bilinear representation is derived by extracting one of the vector roots (unstable in general). This allows us reduce the cubic nonlinearity to a quadratic one. In our approach, the velocity of solitary wave is xed by truncating the Hirota perturbation expansion and it is found in terms of all three roots.Furthermore, Hirota Bilinear Method is also proposed to solve Brusselator reaction model.The simulations of solutions are illustrated for different polynomial roots and parameters as well.