On solutions of a Boussinesq-type equation with amplitude-dependent nonlinearities: the case of biomembranes

TL;DR

This paper investigates a Boussinesq-type wave equation with amplitude-dependent nonlinearities, deriving solitary wave solutions, analyzing their properties, and studying their interactions, with applications to biomembranes.

Contribution

It introduces a modified Boussinesq equation for biomembranes, derives steady solitary wave solutions, and examines their interactions and stability through numerical simulations.

Findings

Solitary wave solutions are derived and characterized.

Wave interactions are inelastic with radiation effects.

Numerical simulations demonstrate wave train formation.

Abstract

Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with amplitude-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson (2005) for describing longitudinal waves in biomembranes and later improved by Engelbrecht et al. (2015) taking into account the microinertia of a biomembrane. The steady solution in the form of a solitary wave is derived and the influence of nonlinear and dispersive terms over a large range of possible sets of coefficients demonstrated. The solutions emerging from arbitrary initial inputs are found using the numerical simulation. The properties of emerging trains of solitary waves waves are analysed. Finally, the interaction of solitary waves which satisfy the governing equation is studied. The interaction process is not fully elastic and after several interactions radiation effects may be significant. This means that for the present case the solitary waves are not solitons in the strict mathematical sense. However, like in other cases known in solid mechanics, such solutions may be conditionally called solitons.