Can Solitons Predict the Future of Traveling Waves?

[somasimple]

Hello all,

I very interested by biological solitons (travelling waves) but my mathematical/physics knowledge is largely out to date.

I post a picture of a biologic soliton below.
Solitons involve maths/physics but I post the subject here?

My asking is the following
1/ if a soliton is effectively a traveling wave, may we say that future points of its curve are already known since they existed in the past of the wave?
see A, B, C examples.
2/ Since a soliton exists at a time t, may we say that it will at a time t+alpha since the curve exists at time t?

 

[Tide]

I'm not sure what you are asking but, yes, a wave will continue to propagate unless something in the medium stops it. A soliton differs in some respects from "ordinary" waves in that it depends on the nonlinear attributes of the medium in which it is formed. The earliest work on solitons with which I am familiar was done by George Airy who observed them as water waves in canals. They maintained their shape and traveled over substantial distances. He was able to keep up them riding on horseback!

 

[somasimple]

Hi,

If a "my" soliton is measurable/observable at point A, may I say that it is mandatory that it will exist at point B, because B is directly "the future" of A and thus A is "the past" of B.

If B exists, it exists because A existed some time before and curve moved in distance.

(A bit confusing? )

 

[Tide]

I think all that you are asking is whether a soliton propagates. Yes, it does!

 

[somasimple]

Tide,
My question is more mathematical!
A soliton is a traveling wave (it is its definition) so it is propagated for sure.

A soliton has a shape (I took a "sine" one).
An external observer of the soliton may see the wave traveling in front of his eyes.
Another observer placed some distance of the primer will see quite the same "wave" a bit later.

The first one begins to see the wave and the second will see these points contained by the curve a bit later.

It means that if the first observer see the whole curve/shape/soliton thus the second observer, placed close to the first one, will mandatory see the whole shape.

We may conclude that a third observer who see the beginning of the shape will automatically see the whole?

 

[somasimple]

I think that we can deduct the incoming shape from its previous places?

 

[Tide]

Generally, a one dimensional soliton will have the mathematical form $f(x - ct)$ where c is the speed of propagation and f represents some envelope function. Therefore, the wave is somewhere, it came from somewhere and will continue on to somewhere else.

 

[somasimple]

tide,

That is the response I was waiting for!

 

[Tide]

I'm glad I could be of help! :)

 

[somasimple]

tide,

Now, if the wave was electrical, can we say that if we could measure two close points, we will see a present curve and already its future?

Normally in my view, it will?

 

[Claude Bile]

A soliton is an attractor state of a nonlinear system, there are a wide variety of initial conditions that will evolve into a given soliton solution.

You cannot infer the history of a soliton by observing it at a time t, because there are many initial states that result in the soliton solution. This is analogous to following the path of a ball in a bowl. While the ball will always end up in the bottom of the bowl, we cannot infer its path from this observation alone.

Claude.

 

[Tide]

Claude,

Those are good points but I believe that somasimple is asking about a soliton that has already formed and he is concerned about its propagation. I am sure he will let us know if that is what he means.

 

[somasimple]

Claude and Tide,
Thanks for the interest about thsi subject.
There is in Nature, a phenomenon which may be considered as an electrical traveling wave and so called, a soliton.
A neuron cell uses effectively a soliton to propagate a message along its axon. I do not think that the shape is really important in this case and it's a major difference with some concerns of Mathematicians. Neuron uses a soliton because it is a solid way to send a message safely.
We know that the message is sent and propagated. My question is thus in these particular conditions, to know if we may deduct past and future events of the wave.

 

[Claude Bile]

Soliton propagation is essentially determined by the properties of the medium it is traveling in. If the medium is changing during propagation, then the shape of the soliton will probably change as well.

To know the history of the solition in detail, you need to know how the medium varies in detail. If you want to know in less detail, you may still need to know how the medium varies in detail, it's hard to say - that is the nature of a nonlinear system.

If there is a sufficient degree of uniformity along the Axon, then it may be possible to infer the history of the soliton, though it is difficult to gauge exactly what your uncertainties will be.

In any case, it sounds like a very interesting topic.

Claude.

 

[somasimple]

Claude,

You're focussed on the shape and its trip.
As I said it, Science is interested by information carried by the soliton, the shape itself. It is of course a very difficult problem to solve and Nature didn't.

Importance was put in the message gaps and not its shape.
Axon is varying in diameter and shape is varying accordingly, speed too.

Major differences between a conventional soliton and the axon one are:

amplification: shape is amplified along a great portion of axon.
addition/sum: solitons may be added to create a third one.

but the major difference is that collision does not let intact the solitons but are vanishing in contrary.
The mechanism of the axon soliton is simple and effective and uses in fact two linked solitons.

http://www.somasimple.com/forums/showthread.php?t=867

 

[somasimple]

Here is two animations of my solitonic point of view about axon/neurons.

http://www.somasimple.com/flash_anims/atomic11.swf
http://www.somasimple.com/flash_anims/atomic12.swf

Of course I removed the ions channels. I wanted only to show the atomic/ionic forces in action.