Exploration of conservation laws, symmetries, and diverse multiform soliton solutions of the Heimburg equation arising in biomembranes.

In this manuscript, Heimburg's model, a mathematical framework for understanding the propagation of electromechanical pulses in biomembranes and nerves is studied. The model interprets the lipid bilayer of the cell membrane to be a substance that undergoes phase transitions. It signifies that the membrane has a nonlinear behavior to electrical perturbations. Through Lie symmetry analysis, the fundamental symmetries of the governing equation are exposed. In addition, Ibragimov's general conservation theorem is used to generate additional conservation laws, ensuring the solutions' physical applicability. Leveraging the recently developed extended direct algebraic technique (EDAM) and Riccati-Bernoulli sub-ODE method, we explore the dynamics of electromechanical pulses in nerves. Numerous soliton solution types are extracted, such as periodic-singular, peakon, bright, dark, kink, combo, and mixed-type solitons. The Hamiltonian characteristic is used to further analyze the stability of these diverse solutions and validate the system's conserved energy features. Furthermore, by utilizing the linear stability approach to study the modulational instability (MI) of a chosen model, offering insights into the circumstances in which perturbations decrease or increase. The resulting two-dimensional, three-dimensional, and contour visualizations of the electromechanical pulse provide valuable insights into nerve functionality, with potential applications in the biological sciences.