An action potential is the firing of electrical signals from one neuron to another caused by a brief change in the voltage across the cell membrane. This is due to the flow of certain ions into and out of the neuron. This Module uses potassium+ (K+) and sodium+(Na+) in most of the models that explain this mechanism.
Alan Hodgkin and Andrew Huxley discovered a Nerve Impulse Flow from a giant squid axon in 1952. They developed a theorem called The Hodgkin-Huxley Model or conductance-based model, a mathematical model that describes how action potentials in neurons are initiated and propagated. It is a set of nonlinear differential equations that approximates the electrical characteristics of excitable cells such as neurons.
There are properties of the mechanics of an action potential that are not explained by the Hodgkin-Huxley Model:


Why nerves display thickness and length variations under the influence of the action potential.


Why the action potential can be excited by a mechanical stimulus.


Why during the first phase of the nerve pulse, heat is released from the membrane, while it is reabsorbed during the second phase.


It seems as if the mechanical and the heat signatures rather indicate that the nerve pulse in an adiabatic and reversible phenomenon such as the propagation of a mechanical wave.


A Soliton is a solitary, self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. The mechanics of an action potential seem to be a similar phenomenon. Therefore, we can use a modified version of the Wave Equation to try to explain the mechanics of the propagation of an an action potential and develop a new model.
An electroencephalogram (EEG) is a test that tracks and records brain wave patterns and it can be used to study neuron action potentials.