| Abstract: |
| We present the original 1D equation for the shape $u(x, t)$ of a nerve pulse [1] in the moving frame as a Boussinesq-like equation $$u_{tt}=\left(u-\frac{p}{2} u^2 + \frac{q}{3} u^3 - h u_{xx}\right)_{xx}$$
and study the influence of the nonlinear coefficients $(p, q)$ on the solution, and also the influence of the $h$ coefficient for the high-order diffusion term $u_{xxxx}$. We explore a large parametric set to study the elastic/inelastic interactions and the velocity/amplitude/phase shifts for localized pulses, solitary waves, critical velocities, newly born excitations, annihilation events, and the cross-modulation effect typical in other Boussinesq or vector Schr\odinger equations. This equation admits the Hamiltonian system
$$u_t = q_{xx}$$
$$q_t= u - \frac{p}{2} u^2 + \frac{q}{3} u^3 - h u_{xx}.$$
We use the analytic 1-soliton solution obtained in [1] to pose a Cauchy problem for this Hamiltonian system. We find the energetic radiation contribution and check its consistency with conserved energy in the asymptotic solutions. We study the energy functional and understand its definiteness depending on $(p, q, h)$ parameters magnitudes. This give us an information about the soliton stability.
$${\rm Bibliography}$$
$${\rm [1] A. Gonzalez-Perez, L. D. Mosgaard, R. Budvytyte, E. Villagran-Vargas, A. D. Jackson, and T. Heimburg, Solitary electromechanical pulses in Lobster neurons, arXiv:1502.07166 (2015). }$$ |
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