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

References

• J. Boussinesq, Théorie de l’intumescence liquide, appelée onde solitaire ou de translation, se propageant dans un canal rectangulaire, C. R. Acad. Sci. 72 (1871), pp. 755–759.
   Google Scholar
• L. Rayleigh, On waves, Philos. Mag. 1 (1876), pp. 257–279.
   Google Scholar
• C.I. Christov, G.A. Maugin, and A.V. Porubov, On Boussinesq’s paradigm in nonlinear wave propagation, C. R. Méc. 335 (2007), pp. 521–535.
   Web of Science ®Google Scholar
• G.A. Maugin, On some generalizations of Boussinesq and KdV systems, Proc. Est. Acad. Sci. Phys. Math. 44 (1995), pp. 40–55.
   Google Scholar
• G.A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford University Press, Oxford, 1999.
   Google Scholar
• A.V. Porubov, Amplification of Nonlinear Strain Waves in Solids, World Scientific, Singapore, 2003.
   Google Scholar
• A. Berezovski, J. Engelbrecht, A. Salupere, K. Tamm, T. Peets, and M. Berezovski, Dispersive waves in microstructured solids, Int. J. Solids Struct. 50 (2013), pp. 1981–1990.
   Web of Science ®Google Scholar
• J. Engelbrecht, K. Tamm, and T. Peets, On mathematical modelling of solitary pulses in cylindrical biomembranes, Biomech. Model. Mechanobiol. 14 (2015), pp. 159–167.
   PubMed Web of Science ®Google Scholar
• J. Engelbrecht, Nonlinear Wave Dynamics. Complexity and Simplicity, Kluwer, Dordrecht, 1997.
   Google Scholar
• R.D. Mindlin, Micro-structure in linear elasticity, Arch. Ration. Mech. Anal. 16 (1964), pp. 51–78.
   Web of Science ®Google Scholar
• G.A. Maugin, Solitons in elastic solids (1938–2010), Mech. Res. Commun. 38 (2011), pp. 341–349.
   Web of Science ®Google Scholar
• J. Engelbrecht, A. Salupere, and K. Tamm, Waves in microstructured solids and the Boussinesq paradigm, Wave Motion 48 (2011), pp. 717–726.
   Web of Science ®Google Scholar
• T. Heimburg and A.D. Jackson, On soliton propagation in biomembranes and nerves, Proc. Natl. Acad. Sci. USA 102 (2005), pp. 9790–9795.
   PubMed Web of Science ®Google Scholar
• S.S.L. Andersen, A.D. Jackson, and T. Heimburg, Towards a thermodynamic theory of nerve pulse propagation, Prog. Neurobiol. 88 (2009), pp. 104–13.
   PubMed Web of Science ®Google Scholar
• R. Appali, U. Van Rienen, and T. Heimburg, A comparison of the Hodgkin--Huxley model and the soliton theory for the action potential in nerves, in Advance Planar Lipid Bilayers Liposomes, Vol. 16, Chapter 9, A. Iglic, ed. Academic Press, Cambridge, 2012, pp. 275–299.
   Google Scholar
• J.K. Mueller and W.J. Tyler, A quantitative overview of biophysical forces impinging on neural function, Phys. Biol. 11 (2014), p. 051001.
   PubMed Web of Science ®Google Scholar
• N. Zabusky and M. Kruskal, Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965), pp. 240–243.
   Web of Science ®Google Scholar
• P. Drazin and R. Johnson, Solitons: And Introduction, Cambridge University Press, Cambridge, 1989.
   Google Scholar
• A. Salupere, P. Peterson, and J. Engelbrecht, Long-time behaviour of soliton ensembles. Part I -- Emergence of ensembles, Chaos Solitons Fractals 14 (2002), pp. 1413–1424.
   Web of Science ®Google Scholar
• D. Debanne, E. Campanac, A. Bialowas, E. Carlier, and G. Alcaraz, Axon physiology, Physiol. Rev. 91 (2011), pp. 555–602.
   PubMed Web of Science ®Google Scholar
• A.L. Hodgkin and A.F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. 117 (1952), pp. 500–544.
   PubMed Web of Science ®Google Scholar
• K. Iwasa, I. Tasaki, and R. Gibbons, Swelling of nerve fibers associated with action potentials, Science 210 (1980), pp. 338–339.
   PubMed Web of Science ®Google Scholar
• I. Tasaki, A macromolecular approach to excitation phenomena: Mechanical and thermal changes in nerve during excitation, Physiol. Chem. Phys. Med. NMR 20 (1988), pp. 251–268.
   PubMed Web of Science ®Google Scholar
• T. Heimburg and A. Jackson, On the action potential as a propagating density pulse and the role of anesthetics, Biophys. Rev. Lett. 2 (2007), pp. 57–78.
   Google Scholar
• F. Maurin and A. Spadoni, Wave propagation in periodic buckled beams. Part I: Analytical models and numerical simulations, Wave Motion 66 (2016), pp. 190–209.
   Web of Science ®Google Scholar
• F. Maurin and A. Spadoni, Wave propagation in periodic buckled beams. Part II: Experiments, Wave Motion 66 (2016), pp. 210–219.
   Web of Science ®Google Scholar
• M.J. Ablowitz, Nonlinear Dispersive Waves. Asymptotic Analysis and Solitons, Cambridge University Press, Cambridge, 2011.
   Google Scholar
• T. Kakutani and N. Yamasaki, Solitary Waves on a Two-layer Fluid, J. Phys. Soc. Jpn. 45 (1978), pp. 674–679.
   Google Scholar
• A.V. Slyunyaev and E.N. Pelinovski, Dynamics of large-amplitude solitons, J. Exp. Theor. Phys. 116 (1999), pp. 173–181.
   Web of Science ®Google Scholar
• T. Peets, K. Tamm, and J. Engelbrecht, On the role of nonlinearities in the Boussinesq-type wave equations, Wave Motion. In press. doi:10.1016/j.wavemoti.2016.04.003.
   Google Scholar
• B. Lautrup, R. Appali, A.D. Jackson, and T. Heimburg, The stability of solitons in biomembranes and nerves, Eur. Phys. J. E. Soft Matter 34 (2011), pp. 1–9.
   Web of Science ®Google Scholar
• T. Peets and K. Tamm, On mechanical aspects of nerve pulse propagation and the Boussinesq paradigm, Proc. Estonian Acad. Sci. 64 (2015), pp. 331–337.
   Web of Science ®Google Scholar
• K. Tamm and T. Peets, On solitary waves in case of amplitude-dependent nonlinearity, Chaos Solitons Fractals 73 (2015), pp. 108–114.
   Web of Science ®Google Scholar
• T. Peets, K. Tamm, and J. Engelbrecht, Numerical investigation of mechanical waves in biomembranes, in Conference Proceeding YIC GACM 2015 3rd ECCOMAS Young Investing Conference 6th GACM Colloquium, July 20--23, 2015, S. Elgeti and J.W. Simon, eds. Aachen, Germany, 2015, pp. 1–4.
   Google Scholar
• B. Fornberg and D.M. Sloan, A review of pseudospectral methods for solving partial differential equations, Acta Numer. 3 (1994), pp. 203–267.
   Google Scholar
• B. Fornberg, A Practical Guide to Pseudospectral Methods, Cambridge University Press, Cambridge, 1998.
   Google Scholar
• A. Salupere, The pseudospectral method and discrete spectral analysis, in Applied Wave Mathematics, E. Quak and T. Soomere, eds. Springer Berlin Heidelberg, Berlin, 2009, pp. 301–334.
   Google Scholar
• E. Jones, T. Oliphant, and P. Peterson, SciPy: Open source scientific tools for Python, (2007). Available at http://www.scipy.org.
   Google Scholar
• A. Hindmarsh, ODEPACK, a Systematized Collection of ODE Solvers, Vol. 1, North-Holland, Amsterdam, 1983.
   Google Scholar
• J. Engelbrecht, K. Tamm, and T. Peets, On solutions of a Boussinesq-type equation with amplitude-dependent nonlinearities: The case of biomembranes, preprint (2016), Available at arXiv:1606.07678 [nlin.PS].
   Google Scholar
• R. Dodd, J. Eilbeck, J. Gibbon, and H. Morris, Solitons and Nonlinear Wave Equations, Academic Press Inc. Ltd., London, 1982.
   Google Scholar
• J. Engelbrecht, Beautiful dynamics, Proc. Estonian Acad. Sci. Phys./Math. 1 (1995), pp. 108–119.
   Google Scholar