Understanding Solitary Waves: Investigating the Motive Behind Their Formation

[somasimple]

Hi,

I'm not really sure the thread fits this forum but I'm not a mathemacian, too.

I posted below a picture of a soliton (solitary wave) and I have some questions about it.
We are seeing two snapshots taken at time t1 and t2.
If t2=t1+a with a, small enough?

Can we say that fb-fa equals a kind like of derivative of f (i.e. f')?
can we say that this difference is the motive/reason of the soliton since fa + (fb-fa) = fb?

 

[HallsofIvy]

If I understand your picture correctly, fa is the height of the wave at a given time, t1, and position,x1, and fb is the height of the wave at a later time, t2, and positon,x2, such that x1-ct1= x2- ct2; in other words, "moving with the wave". Given that, fb-fa= 0. I have no idea what you might mean by "motive/reason" of a soliton.

 

[somasimple]

Hi,

The graph/picture shows two aspects of the same soliton at t1 and t2.
fa is the "curve" at t1 and fb is the curve at t2.

I tried with discrete values (the y values of the curves fa and fb) and I found that substracting fb-fa gives a motive (another curve) that looks like the derivative of fa or fb.
If I add this motive curve values to fa then I obtain fb?
(It seems to work).

 

[HallsofIvy]

If fa and fb are curves, then what do you mean by "fb-fa"? I don't know how to subtract curves! Do you mean that fb is a function of x, fb(x)= f(x,b) where f(x,t) is the height of the curve at t= b? In that case, fb- fa= f(x,b)- f(x,a) and then
lim_{a\rightarrow b}\frac{fb-fa}{b-a}
is the partial derivative of f with respect to t.

I still don't know what a "motive curve" is! I suspect you are translating from some language I don't speak.

 

[somasimple]

Hi,

Sorry for my poor maths language and unfortunately I'm French that complicates our affair.

We could say that fa and fb are the same shape (motive/reason) of the traveling wave taken at diffrent time t1 and t2.
The shape has of course an equation that I do not know but I have discrete values. With these values I can substract 2 digitized curves that gives a third set of values that looks like the derivative of the equation of the shape?

This third set has also an equation but if I consider only the set I have, I can reproduce fb (shape) simply adding the values of fa with the values I got with my previous computation.

Hope it is a bit clearer?