Soliton Solutions in Wave Theory

[Sage Lee]

Hello. I'm not sure the template applies here, as this isn't a textbook style question. I tried to read over the rules and I hope this is the relevant place to put the following query:

I am trying to design a research project in my 1st Engineering Physics class, in which we have full freedom to pick anything we might want to observe, collect data on, and analyze.

Because I'm a moron, I decided to tackle something beyond the scope of this first course, which is of course encouraged. My original thought was simply to "drop different masses into different types of liquid and containers with different depths, and measure 'stuff' [velocity of ripples, height of waves created, etc] and see how that 'stuff' changed as we changed the different variables." That's really the extent of where I'm at in physics so far.

Anyway, today I ran the idea by my teacher, who mentioned that it was a fine idea but that we would probably want to study a particular type of wave, since wave theory has many parts to it. He then suggested Tsunami type waves, and directed me to "Solitons" and "single wave equations".

He's a very helpful teacher, and he would be glad to help further with any questions I have. However, he's a busy man in great demand, so we only got a few minutes of his time, and furthermore, it's now the weekend (classes only through Thursday), so I find myself still with questions that I still need to get answered before I meet with some of my group members tomorrow to start experimenting.

We have a mini-wave tank (meaning ~15 cm deep, ~10cm wide, and 1-1.5 meters long), and the above described ideas. So my question, finally, is this: where do we start here, in an experimental sense? I looked up solitons and single wave equations, but so much of the material is beyond what I can just jump into considering my current understanding of the material. I can't seem to find any "introductory" material relating to what we're trying to do. My team members are all worried we picked something too hard, and they expect me to figure all this out since it was my idea.

So here I am. Can anybody help me out with some direction here? I mean, of course we're going to create waves, but so far all I've come up with is that we might need the sine-gordon equation, and something about "kink solutions," but I haven't the foggiest idea what that really means, or how to go about applying this stuff to our experiments, or even what we should be observing/measuring beyond the stuff above that I mentioned that seems obvious.

If anybody has any suggestions or a good place to start learning some of this material for myself, at a reasonable speed/level for my situation, it would be much appreciated. Thank you in advance for your time.

 

[waveandmatter]

When i once looked up nonlinear wave equations and solitons for recreational purposes i got the feeling that what you are looking for should be the Korteweg- de Vries Equation

http://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation

I don't know much about the topic, but the article does seem understandable and maybe from the math you can get an idea what would have to be done experimentally?

 

[Sage Lee]

Thanks, I checked that out. But it's still a little above me; I have not yet learned how to solve partial differential equations, nor have we gotten into multi-variable calculus yet (learning integrals in Calc 2). In physics, we've basically learned vector math, and have used that to map 2-dimensional motion, and we are just now getting into Energy of systems (PE and KE).

So even that Wikipedia page you linked loses me pretty quickly. It mentions terms like, "the inverse scattering transform," and when I look to see how that works, I just run into more language that is currently beyond my understanding (forward scattering, Lax pair, etc). I just feel that if these concepts were explained to me in simpler terms, it really wouldn't be that hard to understand what's going on - at least enough to make an experiment about it, and talk reasonably intelligently about the things we observe for the eventual 2500 word paper we need to write in a couple weeks. But everything I can find on Wikipedia just looks so damn complicated since I don't already know the lingo and math involved.