Biological Solitons in Biomembranes: Analytical Solutions of a Boussinesq-Type Equation with Amplitude-Dependent Nonlinearity

Abstract

Boussinesq-type equations (BTE) emerge in various fields of fluid and solid mechanics, particularly where nonlinearities and dispersion are considered. Boussinesq-type equations are used to model wave effects in biomembranes, particularly longitudinal waves. They can account for nonlinear and dispersive effects that are important for characteristic wave behavior in biomembranes, composed of lipids, with distinct nonlinear effects. This provides a realistic description of longitudinal mechanical waves in nerve membranes. In this research, we investigate the Boussinesq-type equation that describes the waves in biomembranes with amplitude-dependent nonlinearities, using the Khater method (KM) and the Jacobi elliptic function method (JEFM). In addition to producing generic biological answers, the proposed methods allow the analysis of single wave solutions. These methods make it easier to derive solutions for solitary waves, which occur in a variety of forms, including bell, antibell, periodic, anti-kink and kink solitons. Each of these waves has a wide range of possible applications in biomathematics. Some of the findings are displayed as contour, 2D, and 3D graphics with particular parameter values applied under the specified conditions in order to highlight the important propagation properties. To the best of our knowledge, the biological solitons of the considered model have not been reported by using the proposed techniques in the literature. These results provide new theoretical insights into wave phenomena in biomembranes and may contribute to biological physics and nonlinear science.OPEN ACCESS Received: 04/11/2025 Accepted: 15/12/2025 Published: 23/01/2026

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