The dynamic behaviour of the Boussinesq-type equation governing longitudinal wave propagation in cylindrical biomembranes is analysed by making use of the pseudospectral method. It is shown how the dispersion type has a significant effect on the solution. The effects of other parameters are also considered.
1. Abramson, H. N., Plass, H. J., and Ripperger, E. A. Stress wave propagation in rods and beams. In Advances in Applied Mechanics, Vol. 5 (Dryden, H. and von Karman, T., eds). Academic Press, New York, 1958, 111–194.
http://dx.doi.org/10.1016/s0065-2156(08)70019-x
2. Engelbrecht, J., Salupere, A., and Tamm, K. Waves in microstructured solids and the Boussinesq paradigm. Wave Motion, 2011, 48(8), 717–726.
http://dx.doi.org/10.1016/j.wavemoti.2011.04.001
3. Maugin, G. A. Nonlinear Waves in Elastic Crystals. Oxford University Press, Oxford, 1999.
4. Heimburg, T. and Jackson, A. D. On soliton propagation in biomembranes and nerves. P. Natl. Acad. Sci. USA, 2005, 102(28), 9790–9795.
http://dx.doi.org/10.1073/pnas.0503823102
5. Engelbrecht, J., Tamm, K., and Peets, T. On mathematical modelling of solitary pulses in cylindrical biomembranes. Biomech. Model. Mechanobiol., 2015, 14(1), 159–167.
http://dx.doi.org/10.1007/s10237-014-0596-2
6. Tamm, K. and Peets, T. On solitary waves in case of amplitude-dependent nonlinearity. Chaos Soliton. Fract., 2015, 73, 108–114.
http://dx.doi.org/10.1016/j.chaos.2015.01.013
7. Maugin, G. A. and Christov, C. I. Nonlinear duality between elastic waves and quasi-particles. In Selected Topics in Nonlinear Wave Mechanics (Christov, C. I. and Guran, A., eds). Birkhäuser, Boston, MA, 2002, 101–145.
http://dx.doi.org/10.1007/978-1-4612-0095-6_4
8. Christov, C. I., Maugin, G. A., and Porubov, A. V. On Boussinesq’s paradigm in nonlinear wave propagation. C. R. Mécanique, 2007, 335(9–10), 521–535.
http://dx.doi.org/10.1016/j.crme.2007.08.006
9. Mueller, J. K. and Tyler, W. J. A quantitative overview of biophysical forces impinging on neural function. Phys. Biol., 2014, 11(5), 051001.
http://dx.doi.org/10.1088/1478-3975/11/5/051001
10. Iwasa, K., Tasaki, I., and Gibbons, R. Swelling of nerve fibers associated with action potentials. Science, 1980, 210(4467), 338–339.
http://dx.doi.org/10.1126/science.7423196
11. Tasaki, I. A macromolecular approach to excitation phenomena: mechanical and thermal changes in nerve during excitation. Physiol. Chem. Phys. Med. NMR, 1988, 20, 251–268.
12. Hodgkin, A. L. and Huxley, A. F. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol., 1952, 117, 500–544.
http://dx.doi.org/10.1113/jphysiol.1952.sp004764
13. Nagumo, J., Arimoto, S., and Yoshizawa, S. An active pulse transmission line simulating nerve axon. Proc. IRE, 1962, 50(10), 2061–2070.
http://dx.doi.org/10.1109/jrproc.1962.288235
14. Engelbrecht, J. On theory of pulse transmission in a nerve fibre. P. Roy. Soc. Lond. A Mat., 1981, 375(1761), 195–209.
15. Tasaki, I., Kusano, K., and Byrne, P. M. Rapid mechanical and thermal changes in the garfish olfactory nerve associated with a propagated impulse. Biophys. J., 1989, 55(6), 1033–1040.
http://dx.doi.org/10.1016/S0006-3495(89)82902-9
16. Heimburg, T. and Jackson, A. D. On the action potential as a propagating density pulse and the role of anesthetics. Biophys. Rev. Lett., 2007, 2, 57–78.
http://dx.doi.org/10.1142/S179304800700043X
17. Vargas, E. V., Ludu, A., Hustert, R., Gumrich, P., Jackson, A. D., and Heimburg, T. Periodic solutions and refractory periods in the soliton theory for nerves and the locust femoral nerve. Biophys. Chem., 2011, 153(2–3), 159–167.
http://dx.doi.org/10.1016/j.bpc.2010.11.001
18. Porubov, A. V. and Maugin, G. A. Longitudinal strain solitary waves in presence of cubic non-linearity. Int. J. Nonlinear Mech., 2005, 40, 1041–1048.
http://dx.doi.org/10.1016/j.ijnonlinmec.2005.03.001
19. Slyunyaev, A. V. and Pelinovski, E. N. Dynamics of large-amplitude solitons. J. Exp. Theor. Phys., 1999, 89(1), 173–181.
http://dx.doi.org/10.1134/1.558966
20. Salupere, A. The pseudospectral method and discrete spectral analysis. In Applied Wave Mathematics (Quak, E. and Soomere, T., eds). Springer, Berlin, 2009, 301–334.
http://dx.doi.org/10.1007/978-3-642-00585-5_16
21. Salupere, A., Engelbrecht, J., and Maugin, G. A. Solitonic structures in KdV-based higher-order systems. Wave Motion, 2001, 34(1), 51–61.
http://dx.doi.org/10.1016/S0165-2125(01)00069-5
22. Bona, J. L. and Sachs, R. L. Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. Commun. Math. Phys., 1988, 118, 15–29.
http://dx.doi.org/10.1007/BF01218475