The Lie group and integrability of the Fisher type travelling wave equation (Q844065)

The authors consider reaction-diffusion equations of the form \[ u_t = u_{xx} + u (1-u) (1 + \gamma u) \tag{*} \] with \(\gamma\) a real parameter and \(u (x,t) \in {\mathbb R}\). The travelling wave solutions \(u(x,t) = v (x- c t) \equiv v (z)\) are obtained as solutions to the ODE \[ v'' + c v' + v (1-v) (1 + \gamma v) = 0. \tag{**} \] The paper studies the symmetry properties of (\(**\)), by establishing that it admits a two-parameter Lie group of symmetries provided either \(\{ c^2 = 9/2 , \gamma = 1 \}\) or \(\{ c^2 = 25/6 , \gamma = 0 \}\). Under these conditions, they use a method developed in previous work of themselves to give two independent first integrals which permit that these the corresponding travelling wave solutions can be expressed by elementary functions.