Heimburg and Jackson devised a mathematical model known as the Heimburg model to describe the transmission of electromechanical pulses in nerves, which is a significant step forward. The major objective of this paper was to examine the dynamics of the Heimburg model by extracting closed-form wave solutions. The proposed model was not studied by using analytical techniques. For the first time, innovative analytical solutions were investigated using the exp$\left(-\phi \left(\xi \right)\right)$-expansion method to illustrate the dynamic behavior of the electromechanical pulse in a nerve. This approach generates a wide range of general and broad-spectral solutions with unknown parameters. For the definitive value of these constraints, the well-known periodic- and kink-shaped solitons were recovered. By giving different values to the parameters, the 3D, 2D, and contour forms that constantly modulate in the form of an electromechanical pulse traveling through the axon in the nerve were created. The discovered solutions are innovative, distinct, and useful and might be crucial in medicine and biosciences.

The Riemann wave equation is an intriguing nonlinear equation in the areas of tsunamis and tidal waves in oceans, electromagnetic waves in transmission lines, magnetic and ionic sound radiations in plasmas, static and uniform media, etc. In this innovative research, the analytical solutions of the fractional Riemann wave equation with a conformable derivative were retrieved as a special case, and broad-spectrum solutions with unknown parameters were established with the improved (G’/G)-expansion method. For the various values of these unknown parameters, the renowned periodic, singular, and anti-singular kink-shaped solitons were retrieved. Using the Maple software, we investigated the solutions by drawing the 3D, 2D, and contour plots created to analyze the dynamic behavior of the waves. The discovered solutions might be crucial in the disciplines of science and ocean engineering.