In this paper, the bounded traveling wave solutions of the simplified model of the generalized neural conduction Burgers–Huxley equation with high-order nonlinear terms are investigated. The paper begins with a detailed qualitative analysis of the dynamical system corresponding to the traveling wave solutions of this model, and gives conclusions on the existence conditions and the number of bounded traveling wave solutions of this model. Using the knowledge about the reaction-diffusion equations, the effect of wave speed on the solution is analyzed, we find a threshold value \(c_1\) which can depict the wave speed. By using the analysis method based on the first integral and the method of undetermined assumptions, various difficulties caused by the high-order nonlinearity of the studied equation are overcome, and several important precise impulse solutions of this model are derived, as well as the wavefront solutions that exist when the wave speed is greater than the threshold value. In particular, according to the theory of rotating vector field, the undetermined form of the approximate solutions for oscillatory attenuation solutions formed by the rupture of the homoclinic orbit and the heteroclinic orbit in consequence of the action of small wave speed is properly designed, and the analytic approximate solutions for both kinds of oscillatory attenuation solutions of this model are obtained. Further, by establishing the integral equation between the implicit precise solution and the approximate solution of the oscillatory solution, and investigating the asymptotic behavior of the oscillatory solution at infinity, the error estimates between the implicit precise solution and the approximate solution are given, and the global errors are infinitesimal quantities that decrease exponentially.
