The generalizing riccati equation mapping method's application for detecting soliton solutions in biomembranes and nerves

In this work, we examine the Heimburg model, which describes how electromechanical pulses are transmitted through nerves by using the generalizing Riccati equation mapping method. This approach is regarded as one of the most recent efficient analytical approaches for nonlinear evolution equations, yielding numerous different types of solutions for the model under consideration. We get novel analytic exact solitary wave solutions, including exponential, hyperbolic, and trigonometric functions. These solutions comprises solitary wave, kink, singular kink, periodic, singular soliton, combined dark bright soliton, and breather soliton. To understand the physical principles and significance of the technique the well-furnished results are ultimately displayed in a variety of 2D, 3D, and contour profiles. Additionally, a stability study of the derived solutions is conducted, demonstrating that the steady state is stable under specific parameter restrictions, however the breach of these requirements results in instability due to the exponential increase of perturbations. The results of this work shed light on the importance of studying various nonlinear wave phenomena in nonlinear optics and physics by showing how important it is to understand the behaviour and physical meaning of the studied model. The employed methodology possesses sufficient capability, efficacy, and brevity to enable further research.