In this article, we apply a direct influential approach namely enhanced modified simple equation (EMSE) method to integrate the Burgers–Huxley (BH) and FitzHugh–Nagumo (FHN) equations which explain nerve pulse propagation in nerve fibers, circuit theory as well as wall motion in liquid crystals. The EMSE technique usually used to obtain pulse wave with time variety constants. In fact, we present a set of explicit traveling wave solutions for BH model and three sets of such solutions for FHN model using the technique. Choosing various particular values of the free parameters, we determine kinky periodic pulse, single solitonic pulse, bell type pulse in the kink waves. In particular, we obtain both bright and dark type kink and bell pulse waves from the solutions and interactions between time varying periodic waves with the solitons. We incorporate some 2D and 3D plots to illustrate the wave characteristics (pulses, directions and amplitudes) of the obtained solutions. The solutions of BH and FHN equations can explain nerve pulse propagation in nerve fibers in the population genetics, circuit theory as well as wall motion in liquid crystals.
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