One parameter family of Compacton Solutions in a class of Generalized Korteweg-DeVries Equations

We study the generalized Korteweg-DeVries equations derivable from the Lagrangian: $L(l,p) = \int \left( \frac{1}{2} \varphi_{x} \varphi_{t} - { {(\varphi_{x})^{l}} \over {l(l-1)}} + \alpha(\varphi_{x})^{p} (\varphi_{xx})^{2} \right) dx,$ where the usual fields $u(x,t)$ of the generalized KdV equation are defined by $u(x,t) = \varphi_{x}(x,t)$. For $p$ an arbitrary continuous parameter $0< p \leq 2 ,l=p+2$ we find compacton solutions to these equations which have the feature that their width is independent of the amplitude. This generalizes previous results which considered $p=1,2$. For the exact compactons we find a relation between the energy, mass and velocity of the solitons. We show that this relationship can also be obtained using a variational method based on the principle of least action.Comment: Latex 4 pages and one figure available on reques

Universal scaling and ferroelectric hysteresis regimes in the giant squid axon propagating action potential: a Phase Space Approach

The experimental data for the giant squid axon propagated action potential is examined in phase space. Plots of capacitive and ionic currents vs. potential exhibit linear portions providing temperature dependent time rates and maximum conductance constants for sodium and Potassium channels. First order phase transitions of ionic channels are identified. Incorporation of time rates into Avrami equations for fractions of open channels yields for each channel a temperature independent dimensionless constant that is close in value to the fine structure constant. It also reveals temperature independent scaling exponents. Evidence is presented that the action potential traverses a ferroelectric hysteresis loop. This results in a second order phase transition polarization flip at the peak of the action potential, followed by closing of sodium and opening of potassium channels, and finally closing the loop by reversing the polarization flip as the resting potential is reached. The existence of this hysteresis loop for the giant squid action potential suggests the possibility of neurons with two stable states, the basis for memory storage and retrieval.Comment: 19 pages, 6 figure