Exact solitary-wave solutions of the Burgers–Huxley and Bradley–Harper equations

It is shown that the exact soliton-like solutions of nonlinear wave mechanics evolution equations can be obtained by direct perturbation method based on the solution of a linearized equation. The sought solutions are sums of the perturbation series which can be found using the requirement that the series are to be geometric. This requirement leads to the conditions for the coefficients of the equations and parameters of the sought solutions. The exact solitary-wave solutions of the nonlinear non-integrable Burgers–Huxley equation and the generalized Bradley–Harper equation are obtained. The conditions are formulated under which these solutions have the form of a wave front. It is shown that these solutions can also be found from the system of Riccati equations, that is equivalent to the original equation. By utilizing the Cole–Hopf transformation, the generalized Bradley–Harper equation is reduced to a second-order linear differential equation with constant coefficients.