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

Abstract

Boussinesq-type wave equations involve nonlinearities and dispersion. In this paper a Boussinesq-type equation with displacement-dependent nonlinearities is presented. Such a model was proposed by Heimburg and Jackson for describing longitudinal waves in biomembranes and later improved by Engelbrecht, Tamm and Peets 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 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.

Keywords:

• Elastic waves
• lipids
• mechanical behaviour
• nonlinearities
• dispersion
• solitary waves

Acknowledgements

The valuable comments of the reviewer are acknowledged.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the European Union through the European Regional Development Fund [Estonian Programme TK 124]; the Estonian Research Council [projects IUT 33-24, PUT 434].